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Theorem sltres 31571
Description: If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
sltres ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → 𝐴 <s 𝐵))

Proof of Theorem sltres
Dummy variables 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noreson 31567 . . . . . . 7 ((𝐴 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
213adant2 1078 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
3 noreson 31567 . . . . . . 7 ((𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
433adant1 1077 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
5 sltintdifex 31570 . . . . . . 7 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ V))
6 onintrab 6963 . . . . . . 7 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ V ↔ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On)
75, 6syl6ib 241 . . . . . 6 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On))
82, 4, 7syl2anc 692 . . . . 5 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On))
98imp 445 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On)
10 simpl3 1064 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → 𝑋 ∈ On)
11 sltval2 31563 . . . . . . . . . . . 12 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) ↔ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
122, 4, 11syl2anc 692 . . . . . . . . . . 11 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) ↔ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
13 fvex 6168 . . . . . . . . . . . . 13 ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
14 fvex 6168 . . . . . . . . . . . . 13 ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
1513, 14brtp 31400 . . . . . . . . . . . 12 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
16 1n0 7535 . . . . . . . . . . . . . . . . . 18 1𝑜 ≠ ∅
1716neii 2792 . . . . . . . . . . . . . . . . 17 ¬ 1𝑜 = ∅
18 eqeq1 2625 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 1𝑜 = ∅))
1917, 18mtbiri 317 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
20 ndmfv 6185 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
2119, 20nsyl2 142 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2221adantr 481 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2322orcd 407 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
2421adantr 481 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2524orcd 407 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
26 2on 7528 . . . . . . . . . . . . . . . . . . . . 21 2𝑜 ∈ On
2726elexi 3203 . . . . . . . . . . . . . . . . . . . 20 2𝑜 ∈ V
2827prid2 4275 . . . . . . . . . . . . . . . . . . 19 2𝑜 ∈ {1𝑜, 2𝑜}
2928nosgnn0i 31566 . . . . . . . . . . . . . . . . . 18 ∅ ≠ 2𝑜
3029neii 2792 . . . . . . . . . . . . . . . . 17 ¬ ∅ = 2𝑜
31 eqeq1 2625 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 2𝑜 = ∅))
32 eqcom 2628 . . . . . . . . . . . . . . . . . 18 (2𝑜 = ∅ ↔ ∅ = 2𝑜)
3331, 32syl6bb 276 . . . . . . . . . . . . . . . . 17 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ ∅ = 2𝑜))
3430, 33mtbiri 317 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
35 ndmfv 6185 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
3634, 35nsyl2 142 . . . . . . . . . . . . . . 15 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
3736adantl 482 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
3837olcd 408 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
3923, 25, 383jaoi 1388 . . . . . . . . . . . 12 (((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
4015, 39sylbi 207 . . . . . . . . . . 11 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
4112, 40syl6bi 243 . . . . . . . . . 10 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))))
4241imp 445 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
43 dmres 5388 . . . . . . . . . . . 12 dom (𝐴𝑋) = (𝑋 ∩ dom 𝐴)
4443elin2 3785 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ↔ ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴))
4544simplbi 476 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
46 dmres 5388 . . . . . . . . . . . 12 dom (𝐵𝑋) = (𝑋 ∩ dom 𝐵)
4746elin2 3785 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) ↔ ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵))
4847simplbi 476 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
4945, 48jaoi 394 . . . . . . . . 9 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
5042, 49syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
51 onelss 5735 . . . . . . . 8 (𝑋 ∈ On → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑋))
5210, 50, 51sylc 65 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑋)
5352sselda 3588 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦𝑋)
54 onelon 5717 . . . . . . . . 9 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
559, 54sylan 488 . . . . . . . 8 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
56 intss1 4464 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑦)
57 ontri1 5726 . . . . . . . . . . . . 13 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑦 ↔ ¬ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
5856, 57syl5ib 234 . . . . . . . . . . . 12 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
5958con2d 129 . . . . . . . . . . 11 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → (𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
609, 59sylan 488 . . . . . . . . . 10 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 ∈ On) → (𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6160impancom 456 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → (𝑦 ∈ On → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6255, 61mpd 15 . . . . . . . 8 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})
63 fveq2 6158 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐴𝑋)‘𝑎) = ((𝐴𝑋)‘𝑦))
64 fveq2 6158 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐵𝑋)‘𝑎) = ((𝐵𝑋)‘𝑦))
6563, 64neeq12d 2851 . . . . . . . . . . 11 (𝑎 = 𝑦 → (((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎) ↔ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6665elrab 3351 . . . . . . . . . 10 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ↔ (𝑦 ∈ On ∧ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6766simplbi2 654 . . . . . . . . 9 (𝑦 ∈ On → (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) → 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6867con3d 148 . . . . . . . 8 (𝑦 ∈ On → (¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6955, 62, 68sylc 65 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦))
70 df-ne 2791 . . . . . . . 8 (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) ↔ ¬ ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
7170con2bii 347 . . . . . . 7 (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) ↔ ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦))
7269, 71sylibr 224 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
73 fvres 6174 . . . . . . . 8 (𝑦𝑋 → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
74 fvres 6174 . . . . . . . 8 (𝑦𝑋 → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
7573, 74eqeq12d 2636 . . . . . . 7 (𝑦𝑋 → (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) ↔ (𝐴𝑦) = (𝐵𝑦)))
7675biimpd 219 . . . . . 6 (𝑦𝑋 → (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) → (𝐴𝑦) = (𝐵𝑦)))
7753, 72, 76sylc 65 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → (𝐴𝑦) = (𝐵𝑦))
7877ralrimiva 2962 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦))
79 fvresval 31422 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8079ori 390 . . . . . . . . . . . . . 14 (¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8119, 80nsyl2 142 . . . . . . . . . . . . 13 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
8281eqcomd 2627 . . . . . . . . . . . 12 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
83 eqeq2 2632 . . . . . . . . . . . 12 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜))
8482, 83mpbid 222 . . . . . . . . . . 11 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜)
8584adantr 481 . . . . . . . . . 10 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜)
8685a1i 11 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜))
8721ad2antrl 763 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
8887, 45syl 17 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
89 nofun 31556 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → Fun (𝐵𝑋))
90 fvelrn 6318 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐵𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋))
9190ex 450 . . . . . . . . . . . . . . . . . 18 (Fun (𝐵𝑋) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋)))
9289, 91syl 17 . . . . . . . . . . . . . . . . 17 ((𝐵𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋)))
93 norn 31558 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → ran (𝐵𝑋) ⊆ {1𝑜, 2𝑜})
9493sseld 3587 . . . . . . . . . . . . . . . . 17 ((𝐵𝑋) ∈ No → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
9592, 94syld 47 . . . . . . . . . . . . . . . 16 ((𝐵𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
96 nosgnn0 31565 . . . . . . . . . . . . . . . . 17 ¬ ∅ ∈ {1𝑜, 2𝑜}
97 eleq1 2686 . . . . . . . . . . . . . . . . 17 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
9896, 97mtbiri 317 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
9995, 98nsyli 155 . . . . . . . . . . . . . . 15 ((𝐵𝑋) ∈ No → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
1004, 99syl 17 . . . . . . . . . . . . . 14 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
101100imp 445 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
102101adantrl 751 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
10347simplbi2 654 . . . . . . . . . . . . 13 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
104103con3d 148 . . . . . . . . . . . 12 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → (¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵))
10588, 102, 104sylc 65 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵)
106 ndmfv 6185 . . . . . . . . . . 11 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵 → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
107105, 106syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
108107ex 450 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅))
10986, 108jcad 555 . . . . . . . 8 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)))
110 fvresval 31422 . . . . . . . . . . . . . 14 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
111110ori 390 . . . . . . . . . . . . 13 (¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
11234, 111nsyl2 142 . . . . . . . . . . . 12 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
113112eqcomd 2627 . . . . . . . . . . 11 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
114 eqeq2 2632 . . . . . . . . . . 11 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ((𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))
115113, 114mpbid 222 . . . . . . . . . 10 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)
11684, 115anim12i 589 . . . . . . . . 9 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))
117116a1i 11 . . . . . . . 8 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
11836ad2antll 764 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
119118, 48syl 17 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
120 nofun 31556 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → Fun (𝐴𝑋))
121 fvelrn 6318 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐴𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋))
122121ex 450 . . . . . . . . . . . . . . . . . 18 (Fun (𝐴𝑋) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋)))
123120, 122syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋)))
124 norn 31558 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → ran (𝐴𝑋) ⊆ {1𝑜, 2𝑜})
125124sseld 3587 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋) ∈ No → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
126123, 125syld 47 . . . . . . . . . . . . . . . 16 ((𝐴𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
127 eleq1 2686 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
12896, 127mtbiri 317 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
129126, 128nsyli 155 . . . . . . . . . . . . . . 15 ((𝐴𝑋) ∈ No → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)))
1302, 129syl 17 . . . . . . . . . . . . . 14 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)))
131130imp 445 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
132131adantrr 752 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
13344simplbi2 654 . . . . . . . . . . . . 13 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)))
134133con3d 148 . . . . . . . . . . . 12 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → (¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴))
135119, 132, 134sylc 65 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴)
136135ex 450 . . . . . . . . . 10 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴))
137 ndmfv 6185 . . . . . . . . . 10 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
138136, 137syl6 35 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅))
139115adantl 482 . . . . . . . . . 10 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)
140139a1i 11 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))
141138, 140jcad 555 . . . . . . . 8 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
142109, 117, 1413orim123d 1404 . . . . . . 7 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → (((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))))
143 fvex 6168 . . . . . . . 8 (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
144 fvex 6168 . . . . . . . 8 (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
145143, 144brtp 31400 . . . . . . 7 ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ (((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
146142, 15, 1453imtr4g 285 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
14712, 146sylbid 230 . . . . 5 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
148147imp 445 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
149 raleq 3131 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦)))
150 fveq2 6158 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
151 fveq2 6158 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
152150, 151breq12d 4636 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ((𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥) ↔ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
153149, 152anbi12d 746 . . . . 5 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)) ↔ (∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))))
154153rspcev 3299 . . . 4 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ (∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
1559, 78, 148, 154syl12anc 1321 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
156 sltval 31554 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
1571563adant3 1079 . . . 4 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
158157adantr 481 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
159155, 158mpbird 247 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → 𝐴 <s 𝐵)
160159ex 450 1 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → 𝐴 <s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  wss 3560  c0 3897  {cpr 4157  {ctp 4159  cop 4161   cint 4447   class class class wbr 4623  dom cdm 5084  ran crn 5085  cres 5086  Oncon0 5692  Fun wfun 5851  cfv 5857  1𝑜c1o 7513  2𝑜c2o 7514   No csur 31547   <s cslt 31548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-1o 7520  df-2o 7521  df-no 31550  df-slt 31551
This theorem is referenced by:  sltgtres  31623
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