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Theorem sltval2 27540
Description: Alternate expression for surreal less-than. Two surreals obey surreal less-than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)
Assertion
Ref Expression
sltval2 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem sltval2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sltval 27531 . 2 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
2 fvex 6897 . . . . . . . . . . . . 13 (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
3 fvex 6897 . . . . . . . . . . . . 13 (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
42, 3brtp 5516 . . . . . . . . . . . 12 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
5 1n0 8486 . . . . . . . . . . . . . . . . 17 1o ≠ ∅
65neii 2936 . . . . . . . . . . . . . . . 16 ¬ 1o = ∅
7 eqeq1 2730 . . . . . . . . . . . . . . . 16 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ↔ 1o = ∅))
86, 7mtbiri 327 . . . . . . . . . . . . . . 15 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o → ¬ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
9 fvprc 6876 . . . . . . . . . . . . . . 15 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
108, 9nsyl2 141 . . . . . . . . . . . . . 14 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
1110adantr 480 . . . . . . . . . . . . 13 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
1210adantr 480 . . . . . . . . . . . . 13 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
13 2on0 8480 . . . . . . . . . . . . . . . . 17 2o ≠ ∅
1413neii 2936 . . . . . . . . . . . . . . . 16 ¬ 2o = ∅
15 eqeq1 2730 . . . . . . . . . . . . . . . 16 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o → ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ↔ 2o = ∅))
1614, 15mtbiri 327 . . . . . . . . . . . . . . 15 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
17 fvprc 6876 . . . . . . . . . . . . . . 15 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V → (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
1816, 17nsyl2 141 . . . . . . . . . . . . . 14 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
1918adantl 481 . . . . . . . . . . . . 13 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
2011, 12, 193jaoi 1424 . . . . . . . . . . . 12 ((((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
214, 20sylbi 216 . . . . . . . . . . 11 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
22 onintrab 7780 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V ↔ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
2321, 22sylib 217 . . . . . . . . . 10 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
2423adantl 481 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
25 onelon 6382 . . . . . . . . . . . . . 14 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝑦 ∈ On)
2625expcom 413 . . . . . . . . . . . . 13 (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → 𝑦 ∈ On))
2724, 26syl5 34 . . . . . . . . . . . 12 (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → 𝑦 ∈ On))
28 fveq2 6884 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → (𝐴𝑎) = (𝐴𝑦))
29 fveq2 6884 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → (𝐵𝑎) = (𝐵𝑦))
3028, 29neeq12d 2996 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴𝑦) ≠ (𝐵𝑦)))
3130onnminsb 7783 . . . . . . . . . . . . 13 (𝑦 ∈ On → (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ (𝐴𝑦) ≠ (𝐵𝑦)))
3231com12 32 . . . . . . . . . . . 12 (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝑦 ∈ On → ¬ (𝐴𝑦) ≠ (𝐵𝑦)))
3327, 32syldc 48 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ (𝐴𝑦) ≠ (𝐵𝑦)))
34 df-ne 2935 . . . . . . . . . . . 12 ((𝐴𝑦) ≠ (𝐵𝑦) ↔ ¬ (𝐴𝑦) = (𝐵𝑦))
3534con2bii 357 . . . . . . . . . . 11 ((𝐴𝑦) = (𝐵𝑦) ↔ ¬ (𝐴𝑦) ≠ (𝐵𝑦))
3633, 35imbitrrdi 251 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑦) = (𝐵𝑦)))
3736ralrimiv 3139 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦))
3824, 37jca 511 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦)))
3938ex 412 . . . . . . 7 ((𝐴 No 𝐵 No ) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦))))
4039impac 552 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
41 anass 468 . . . . . 6 ((( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) ↔ ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))))
4240, 41sylib 217 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))))
43 raleq 3316 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦)))
44 fveq2 6884 . . . . . . . 8 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
45 fveq2 6884 . . . . . . . 8 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
4644, 45breq12d 5154 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
4743, 46anbi12d 630 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))))
4847rspcev 3606 . . . . 5 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
4942, 48syl 17 . . . 4 (((𝐴 No 𝐵 No ) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
5049ex 412 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
51 eqeq12 2743 . . . . . . . . . . . . . 14 (((𝐴𝑥) = 1o ∧ (𝐵𝑥) = ∅) → ((𝐴𝑥) = (𝐵𝑥) ↔ 1o = ∅))
526, 51mtbiri 327 . . . . . . . . . . . . 13 (((𝐴𝑥) = 1o ∧ (𝐵𝑥) = ∅) → ¬ (𝐴𝑥) = (𝐵𝑥))
53 1on 8476 . . . . . . . . . . . . . . . . 17 1o ∈ On
54 0elon 6411 . . . . . . . . . . . . . . . . 17 ∅ ∈ On
55 suc11 6464 . . . . . . . . . . . . . . . . . 18 ((1o ∈ On ∧ ∅ ∈ On) → (suc 1o = suc ∅ ↔ 1o = ∅))
5655necon3bid 2979 . . . . . . . . . . . . . . . . 17 ((1o ∈ On ∧ ∅ ∈ On) → (suc 1o ≠ suc ∅ ↔ 1o ≠ ∅))
5753, 54, 56mp2an 689 . . . . . . . . . . . . . . . 16 (suc 1o ≠ suc ∅ ↔ 1o ≠ ∅)
585, 57mpbir 230 . . . . . . . . . . . . . . 15 suc 1o ≠ suc ∅
59 df-2o 8465 . . . . . . . . . . . . . . . 16 2o = suc 1o
60 df-1o 8464 . . . . . . . . . . . . . . . 16 1o = suc ∅
6159, 60eqeq12i 2744 . . . . . . . . . . . . . . 15 (2o = 1o ↔ suc 1o = suc ∅)
6258, 61nemtbir 3032 . . . . . . . . . . . . . 14 ¬ 2o = 1o
63 eqeq12 2743 . . . . . . . . . . . . . . 15 (((𝐴𝑥) = 1o ∧ (𝐵𝑥) = 2o) → ((𝐴𝑥) = (𝐵𝑥) ↔ 1o = 2o))
64 eqcom 2733 . . . . . . . . . . . . . . 15 (1o = 2o ↔ 2o = 1o)
6563, 64bitrdi 287 . . . . . . . . . . . . . 14 (((𝐴𝑥) = 1o ∧ (𝐵𝑥) = 2o) → ((𝐴𝑥) = (𝐵𝑥) ↔ 2o = 1o))
6662, 65mtbiri 327 . . . . . . . . . . . . 13 (((𝐴𝑥) = 1o ∧ (𝐵𝑥) = 2o) → ¬ (𝐴𝑥) = (𝐵𝑥))
6713nesymi 2992 . . . . . . . . . . . . . 14 ¬ ∅ = 2o
68 eqeq12 2743 . . . . . . . . . . . . . 14 (((𝐴𝑥) = ∅ ∧ (𝐵𝑥) = 2o) → ((𝐴𝑥) = (𝐵𝑥) ↔ ∅ = 2o))
6967, 68mtbiri 327 . . . . . . . . . . . . 13 (((𝐴𝑥) = ∅ ∧ (𝐵𝑥) = 2o) → ¬ (𝐴𝑥) = (𝐵𝑥))
7052, 66, 693jaoi 1424 . . . . . . . . . . . 12 ((((𝐴𝑥) = 1o ∧ (𝐵𝑥) = ∅) ∨ ((𝐴𝑥) = 1o ∧ (𝐵𝑥) = 2o) ∨ ((𝐴𝑥) = ∅ ∧ (𝐵𝑥) = 2o)) → ¬ (𝐴𝑥) = (𝐵𝑥))
71 fvex 6897 . . . . . . . . . . . . 13 (𝐴𝑥) ∈ V
72 fvex 6897 . . . . . . . . . . . . 13 (𝐵𝑥) ∈ V
7371, 72brtp 5516 . . . . . . . . . . . 12 ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (((𝐴𝑥) = 1o ∧ (𝐵𝑥) = ∅) ∨ ((𝐴𝑥) = 1o ∧ (𝐵𝑥) = 2o) ∨ ((𝐴𝑥) = ∅ ∧ (𝐵𝑥) = 2o)))
74 df-ne 2935 . . . . . . . . . . . 12 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ (𝐴𝑥) = (𝐵𝑥))
7570, 73, 743imtr4i 292 . . . . . . . . . . 11 ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) → (𝐴𝑥) ≠ (𝐵𝑥))
76 fveq2 6884 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝐴𝑎) = (𝐴𝑥))
77 fveq2 6884 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝐵𝑎) = (𝐵𝑥))
7876, 77neeq12d 2996 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴𝑥) ≠ (𝐵𝑥)))
7978elrab 3678 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ (𝑥 ∈ On ∧ (𝐴𝑥) ≠ (𝐵𝑥)))
8079biimpri 227 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ (𝐴𝑥) ≠ (𝐵𝑥)) → 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
8180adantlr 712 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴𝑥) ≠ (𝐵𝑥)) → 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
82 ssrab2 4072 . . . . . . . . . . . . . . . . . 18 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ On
83 ne0i 4329 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ≠ ∅)
8483adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ≠ ∅)
85 onint 7774 . . . . . . . . . . . . . . . . . 18 (({𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ On ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ≠ ∅) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
8682, 84, 85sylancr 586 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
87 nfrab1 3445 . . . . . . . . . . . . . . . . . . . 20 𝑎{𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}
8887nfint 4953 . . . . . . . . . . . . . . . . . . 19 𝑎 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}
89 nfcv 2897 . . . . . . . . . . . . . . . . . . 19 𝑎On
90 nfcv 2897 . . . . . . . . . . . . . . . . . . . . 21 𝑎𝐴
9190, 88nffv 6894 . . . . . . . . . . . . . . . . . . . 20 𝑎(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
92 nfcv 2897 . . . . . . . . . . . . . . . . . . . . 21 𝑎𝐵
9392, 88nffv 6894 . . . . . . . . . . . . . . . . . . . 20 𝑎(𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
9491, 93nfne 3037 . . . . . . . . . . . . . . . . . . 19 𝑎(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ≠ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
95 fveq2 6884 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑎) = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
96 fveq2 6884 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐵𝑎) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
9795, 96neeq12d 2996 . . . . . . . . . . . . . . . . . . 19 (𝑎 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ≠ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
9888, 89, 94, 97elrabf 3674 . . . . . . . . . . . . . . . . . 18 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ≠ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
9998simprbi 496 . . . . . . . . . . . . . . . . 17 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ≠ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
10086, 99syl 17 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ≠ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
101 df-ne 2935 . . . . . . . . . . . . . . . 16 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ≠ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ ¬ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
102100, 101sylib 217 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ¬ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
103 fveq2 6884 . . . . . . . . . . . . . . . . . 18 (𝑦 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑦) = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
104 fveq2 6884 . . . . . . . . . . . . . . . . . 18 (𝑦 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐵𝑦) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
105103, 104eqeq12d 2742 . . . . . . . . . . . . . . . . 17 (𝑦 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴𝑦) = (𝐵𝑦) ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
106105rspccv 3603 . . . . . . . . . . . . . . . 16 (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ 𝑥 → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
107106ad2antlr 724 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ 𝑥 → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
108102, 107mtod 197 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ 𝑥)
109 simpll 764 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝑥 ∈ On)
110 oninton 7779 . . . . . . . . . . . . . . . . 17 (({𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ On ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ≠ ∅) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
11182, 83, 110sylancr 586 . . . . . . . . . . . . . . . 16 (𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
112111adantl 481 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
113 ontri1 6391 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On) → (𝑥 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ 𝑥))
114109, 112, 113syl2anc 583 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝑥 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ 𝑥))
115108, 114mpbird 257 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝑥 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
116 intss1 4960 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝑥)
117116adantl 481 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝑥)
118115, 117eqssd 3994 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ 𝑥 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
11981, 118syldan 590 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴𝑥) ≠ (𝐵𝑥)) → 𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
12075, 119sylan2 592 . . . . . . . . . 10 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) → 𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
121120fveq2d 6888 . . . . . . . . 9 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
122120fveq2d 6888 . . . . . . . . 9 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
123121, 122breq12d 5154 . . . . . . . 8 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
124123biimpd 228 . . . . . . 7 (((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
125124ex 412 . . . . . 6 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))))
126125pm2.43d 53 . . . . 5 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)) → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
127126expimpd 453 . . . 4 (𝑥 ∈ On → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
128127rexlimiv 3142 . . 3 (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
12950, 128impbid1 224 . 2 ((𝐴 No 𝐵 No ) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
1301, 129bitr4d 282 1 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3o 1083   = wceq 1533  wcel 2098  wne 2934  wral 3055  wrex 3064  {crab 3426  Vcvv 3468  wss 3943  c0 4317  {ctp 4627  cop 4629   cint 4943   class class class wbr 5141  Oncon0 6357  suc csuc 6359  cfv 6536  1oc1o 8457  2oc2o 8458   No csur 27524   <s cslt 27525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fv 6544  df-1o 8464  df-2o 8465  df-slt 27528
This theorem is referenced by:  sltintdifex  27545  sltres  27546  noextendlt  27553  noextendgt  27554  nosepnelem  27563  nosep1o  27565  nosep2o  27566  nosepdmlem  27567  nodenselem8  27575  nosupbnd2lem1  27599  noinfbnd2lem1  27614
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