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Theorem sltres 30864
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 30860 . . . . . . 7 ((𝐴 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
213adant2 1072 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
3 noreson 30860 . . . . . . 7 ((𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
433adant1 1071 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
5 sltintdifex 30863 . . . . . . 7 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ V))
6 onintrab 6867 . . . . . . 7 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ V ↔ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On)
75, 6syl6ib 239 . . . . . 6 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On))
82, 4, 7syl2anc 690 . . . . 5 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On))
98imp 443 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On)
10 simpl3 1058 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → 𝑋 ∈ On)
11 sltval2 30856 . . . . . . . . . . . 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 690 . . . . . . . . . . 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 6095 . . . . . . . . . . . . 13 ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
14 fvex 6095 . . . . . . . . . . . . 13 ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
1513, 14brtp 30695 . . . . . . . . . . . 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 7436 . . . . . . . . . . . . . . . . . 18 1𝑜 ≠ ∅
1716neii 2780 . . . . . . . . . . . . . . . . 17 ¬ 1𝑜 = ∅
18 eqeq1 2610 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 1𝑜 = ∅))
1917, 18mtbiri 315 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
20 ndmfv 6110 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
2119, 20nsyl2 140 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2221adantr 479 . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . 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 7429 . . . . . . . . . . . . . . . . . . . . 21 2𝑜 ∈ On
2726elexi 3182 . . . . . . . . . . . . . . . . . . . 20 2𝑜 ∈ V
2827prid2 4238 . . . . . . . . . . . . . . . . . . 19 2𝑜 ∈ {1𝑜, 2𝑜}
2928nosgnn0i 30859 . . . . . . . . . . . . . . . . . 18 ∅ ≠ 2𝑜
3029neii 2780 . . . . . . . . . . . . . . . . 17 ¬ ∅ = 2𝑜
31 eqeq1 2610 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 2𝑜 = ∅))
32 eqcom 2613 . . . . . . . . . . . . . . . . . 18 (2𝑜 = ∅ ↔ ∅ = 2𝑜)
3331, 32syl6bb 274 . . . . . . . . . . . . . . . . 17 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ ∅ = 2𝑜))
3430, 33mtbiri 315 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
35 ndmfv 6110 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
3634, 35nsyl2 140 . . . . . . . . . . . . . . 15 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
3736adantl 480 . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 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 1382 . . . . . . . . . . . 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 205 . . . . . . . . . . 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 241 . . . . . . . . . 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 443 . . . . . . . . 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 5323 . . . . . . . . . . . 12 dom (𝐴𝑋) = (𝑋 ∩ dom 𝐴)
4443elin2 3759 . . . . . . . . . . 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 474 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
46 dmres 5323 . . . . . . . . . . . 12 dom (𝐵𝑋) = (𝑋 ∩ dom 𝐵)
4746elin2 3759 . . . . . . . . . . 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 474 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
4945, 48jaoi 392 . . . . . . . . 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 5666 . . . . . . . 8 (𝑋 ∈ On → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑋))
5210, 50, 51sylc 62 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑋)
5352sselda 3564 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦𝑋)
54 onelon 5648 . . . . . . . . 9 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
559, 54sylan 486 . . . . . . . 8 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
56 intss1 4418 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑦)
57 ontri1 5657 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 127 . . . . . . . . . . 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 486 . . . . . . . . . 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 454 . . . . . . . . 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 6085 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐴𝑋)‘𝑎) = ((𝐴𝑋)‘𝑦))
64 fveq2 6085 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐵𝑋)‘𝑎) = ((𝐵𝑋)‘𝑦))
6563, 64neeq12d 2839 . . . . . . . . . . 11 (𝑎 = 𝑦 → (((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎) ↔ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6665elrab 3327 . . . . . . . . . 10 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ↔ (𝑦 ∈ On ∧ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6766simplbi2 652 . . . . . . . . 9 (𝑦 ∈ On → (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) → 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6867con3d 146 . . . . . . . 8 (𝑦 ∈ On → (¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6955, 62, 68sylc 62 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦))
70 df-ne 2778 . . . . . . . 8 (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) ↔ ¬ ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
7170con2bii 345 . . . . . . 7 (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) ↔ ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦))
7269, 71sylibr 222 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
73 fvres 6099 . . . . . . . 8 (𝑦𝑋 → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
74 fvres 6099 . . . . . . . 8 (𝑦𝑋 → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
7573, 74eqeq12d 2621 . . . . . . 7 (𝑦𝑋 → (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) ↔ (𝐴𝑦) = (𝐵𝑦)))
7675biimpd 217 . . . . . 6 (𝑦𝑋 → (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) → (𝐴𝑦) = (𝐵𝑦)))
7753, 72, 76sylc 62 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → (𝐴𝑦) = (𝐵𝑦))
7877ralrimiva 2945 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦))
79 fvresval 30714 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8079ori 388 . . . . . . . . . . . . . 14 (¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8119, 80nsyl2 140 . . . . . . . . . . . . 13 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
8281eqcomd 2612 . . . . . . . . . . . 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 2617 . . . . . . . . . . . 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 220 . . . . . . . . . . 11 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜)
8584adantr 479 . . . . . . . . . 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 759 . . . . . . . . . . . . 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 30849 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → Fun (𝐵𝑋))
90 fvelrn 6242 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐵𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋))
9190ex 448 . . . . . . . . . . . . . . . . . 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 30851 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → ran (𝐵𝑋) ⊆ {1𝑜, 2𝑜})
9493sseld 3563 . . . . . . . . . . . . . . . . 17 ((𝐵𝑋) ∈ No → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
9592, 94syld 45 . . . . . . . . . . . . . . . 16 ((𝐵𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
96 nosgnn0 30858 . . . . . . . . . . . . . . . . 17 ¬ ∅ ∈ {1𝑜, 2𝑜}
97 eleq1 2672 . . . . . . . . . . . . . . . . 17 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
9896, 97mtbiri 315 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
9995, 98nsyli 153 . . . . . . . . . . . . . . 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 443 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
102101adantrl 747 . . . . . . . . . . . 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 652 . . . . . . . . . . . . 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 146 . . . . . . . . . . . 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 62 . . . . . . . . . . 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 6110 . . . . . . . . . . 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 448 . . . . . . . . 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 553 . . . . . . . 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 30714 . . . . . . . . . . . . . 14 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
111110ori 388 . . . . . . . . . . . . 13 (¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
11234, 111nsyl2 140 . . . . . . . . . . . 12 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
113112eqcomd 2612 . . . . . . . . . . 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 2617 . . . . . . . . . . 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 220 . . . . . . . . . 10 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)
11684, 115anim12i 587 . . . . . . . . 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 760 . . . . . . . . . . . . 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 30849 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → Fun (𝐴𝑋))
121 fvelrn 6242 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐴𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋))
122121ex 448 . . . . . . . . . . . . . . . . . 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 30851 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → ran (𝐴𝑋) ⊆ {1𝑜, 2𝑜})
125124sseld 3563 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋) ∈ No → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
126123, 125syld 45 . . . . . . . . . . . . . . . 16 ((𝐴𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
127 eleq1 2672 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
12896, 127mtbiri 315 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
129126, 128nsyli 153 . . . . . . . . . . . . . . 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 443 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
132131adantrr 748 . . . . . . . . . . . 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 652 . . . . . . . . . . . . 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 146 . . . . . . . . . . . 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 62 . . . . . . . . . . 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 448 . . . . . . . . . 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 6110 . . . . . . . . . 10 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
138136, 137syl6 34 . . . . . . . . 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 480 . . . . . . . . . 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 553 . . . . . . . 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 1398 . . . . . . 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 6095 . . . . . . . 8 (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
144 fvex 6095 . . . . . . . 8 (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
145143, 144brtp 30695 . . . . . . 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 283 . . . . . 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 228 . . . . 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 443 . . . 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 3111 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦)))
150 fveq2 6085 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
151 fveq2 6085 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
152150, 151breq12d 4587 . . . . . 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 742 . . . . 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 3278 . . . 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 1315 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
156 sltval 30847 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
1571563adant3 1073 . . . 4 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
158157adantr 479 . . 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 245 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → 𝐴 <s 𝐵)
160159ex 448 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 194  wo 381  wa 382  w3o 1029  w3a 1030   = wceq 1474  wcel 1976  wne 2776  wral 2892  wrex 2893  {crab 2896  Vcvv 3169  wss 3536  c0 3870  {cpr 4123  {ctp 4125  cop 4127   cint 4401   class class class wbr 4574  dom cdm 5025  ran crn 5026  cres 5027  Oncon0 5623  Fun wfun 5781  cfv 5787  1𝑜c1o 7414  2𝑜c2o 7415   No csur 30840   <s cslt 30841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-ord 5626  df-on 5627  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-1o 7421  df-2o 7422  df-no 30843  df-slt 30844
This theorem is referenced by: (None)
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