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Theorem dfon2lem5 31390
Description: Lemma for dfon2 31395. Two sets satisfying the new definition also satisfy trichotomy with respect to . (Contributed by Scott Fenton, 25-Feb-2011.)
Hypotheses
Ref Expression
dfon2lem5.1 𝐴 ∈ V
dfon2lem5.2 𝐵 ∈ V
Assertion
Ref Expression
dfon2lem5 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem dfon2lem5
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfon2lem5.1 . . . 4 𝐴 ∈ V
2 dfon2lem5.2 . . . 4 𝐵 ∈ V
31, 2dfon2lem4 31389 . . 3 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐵𝐴))
4 dfpss2 3670 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
5 dfpss2 3670 . . . . . . 7 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵 = 𝐴))
6 eqcom 2628 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
76notbii 310 . . . . . . . 8 𝐵 = 𝐴 ↔ ¬ 𝐴 = 𝐵)
87anbi2i 729 . . . . . . 7 ((𝐵𝐴 ∧ ¬ 𝐵 = 𝐴) ↔ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵))
95, 8bitri 264 . . . . . 6 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵))
104, 9orbi12i 543 . . . . 5 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵)))
11 andir 911 . . . . 5 (((𝐴𝐵𝐵𝐴) ∧ ¬ 𝐴 = 𝐵) ↔ ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵)))
1210, 11bitr4i 267 . . . 4 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∧ ¬ 𝐴 = 𝐵))
13 orcom 402 . . . . 5 ((𝐴𝐵𝐵𝐴) ↔ (𝐵𝐴𝐴𝐵))
14 dfon2lem3 31388 . . . . . . . . 9 (𝐵 ∈ V → (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → (Tr 𝐵 ∧ ∀𝑧𝐵 ¬ 𝑧𝑧)))
152, 14ax-mp 5 . . . . . . . 8 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → (Tr 𝐵 ∧ ∀𝑧𝐵 ¬ 𝑧𝑧))
1615simpld 475 . . . . . . 7 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → Tr 𝐵)
17 psseq1 3672 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
18 treq 4718 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (Tr 𝑥 ↔ Tr 𝐵))
1917, 18anbi12d 746 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝑥𝐴 ∧ Tr 𝑥) ↔ (𝐵𝐴 ∧ Tr 𝐵)))
20 eleq1 2686 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
2119, 20imbi12d 334 . . . . . . . . . 10 (𝑥 = 𝐵 → (((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ↔ ((𝐵𝐴 ∧ Tr 𝐵) → 𝐵𝐴)))
222, 21spcv 3285 . . . . . . . . 9 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → ((𝐵𝐴 ∧ Tr 𝐵) → 𝐵𝐴))
2322expcomd 454 . . . . . . . 8 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐵 → (𝐵𝐴𝐵𝐴)))
2423imp 445 . . . . . . 7 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ Tr 𝐵) → (𝐵𝐴𝐵𝐴))
2516, 24sylan2 491 . . . . . 6 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐵𝐴𝐵𝐴))
26 dfon2lem3 31388 . . . . . . . . 9 (𝐴 ∈ V → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))
271, 26ax-mp 5 . . . . . . . 8 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧))
2827simpld 475 . . . . . . 7 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → Tr 𝐴)
29 psseq1 3672 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
30 treq 4718 . . . . . . . . . . 11 (𝑦 = 𝐴 → (Tr 𝑦 ↔ Tr 𝐴))
3129, 30anbi12d 746 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦𝐵 ∧ Tr 𝑦) ↔ (𝐴𝐵 ∧ Tr 𝐴)))
32 eleq1 2686 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
3331, 32imbi12d 334 . . . . . . . . 9 (𝑦 = 𝐴 → (((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) ↔ ((𝐴𝐵 ∧ Tr 𝐴) → 𝐴𝐵)))
341, 33spcv 3285 . . . . . . . 8 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → ((𝐴𝐵 ∧ Tr 𝐴) → 𝐴𝐵))
3534expcomd 454 . . . . . . 7 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → (Tr 𝐴 → (𝐴𝐵𝐴𝐵)))
3628, 35mpan9 486 . . . . . 6 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐴𝐵))
3725, 36orim12d 882 . . . . 5 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐵𝐴𝐴𝐵) → (𝐵𝐴𝐴𝐵)))
3813, 37syl5bi 232 . . . 4 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐴𝐵𝐵𝐴) → (𝐵𝐴𝐴𝐵)))
3912, 38syl5bir 233 . . 3 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (((𝐴𝐵𝐵𝐴) ∧ ¬ 𝐴 = 𝐵) → (𝐵𝐴𝐴𝐵)))
403, 39mpand 710 . 2 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
41 3orrot 1042 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (𝐴 = 𝐵𝐵𝐴𝐴𝐵))
42 3orass 1039 . . . 4 ((𝐴 = 𝐵𝐵𝐴𝐴𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐵𝐴𝐴𝐵)))
43 df-or 385 . . . 4 ((𝐴 = 𝐵 ∨ (𝐵𝐴𝐴𝐵)) ↔ (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
4442, 43bitri 264 . . 3 ((𝐴 = 𝐵𝐵𝐴𝐴𝐵) ↔ (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
4541, 44bitri 264 . 2 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
4640, 45sylibr 224 1 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3o 1035  wal 1478   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555  wpss 3556  Tr wtr 4712
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-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151  df-uni 4403  df-iun 4487  df-tr 4713  df-suc 5688
This theorem is referenced by:  dfon2lem6  31391  dfon2  31395
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