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Theorem dfon2lem5 33145
Description: Lemma for dfon2 33150. 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 33144 . . 3 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐵𝐴))
4 dfpss2 4013 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
5 dfpss2 4013 . . . . . . 7 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵 = 𝐴))
6 eqcom 2805 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
76notbii 323 . . . . . . . 8 𝐵 = 𝐴 ↔ ¬ 𝐴 = 𝐵)
87anbi2i 625 . . . . . . 7 ((𝐵𝐴 ∧ ¬ 𝐵 = 𝐴) ↔ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵))
95, 8bitri 278 . . . . . 6 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵))
104, 9orbi12i 912 . . . . 5 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵)))
11 andir 1006 . . . . 5 (((𝐴𝐵𝐵𝐴) ∧ ¬ 𝐴 = 𝐵) ↔ ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵)))
1210, 11bitr4i 281 . . . 4 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∧ ¬ 𝐴 = 𝐵))
13 orcom 867 . . . . 5 ((𝐴𝐵𝐵𝐴) ↔ (𝐵𝐴𝐴𝐵))
14 dfon2lem3 33143 . . . . . . . . 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 498 . . . . . . 7 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → Tr 𝐵)
17 psseq1 4015 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
18 treq 5142 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (Tr 𝑥 ↔ Tr 𝐵))
1917, 18anbi12d 633 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝑥𝐴 ∧ Tr 𝑥) ↔ (𝐵𝐴 ∧ Tr 𝐵)))
20 eleq1 2877 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
2119, 20imbi12d 348 . . . . . . . . . 10 (𝑥 = 𝐵 → (((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ↔ ((𝐵𝐴 ∧ Tr 𝐵) → 𝐵𝐴)))
222, 21spcv 3554 . . . . . . . . 9 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → ((𝐵𝐴 ∧ Tr 𝐵) → 𝐵𝐴))
2322expcomd 420 . . . . . . . 8 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐵 → (𝐵𝐴𝐵𝐴)))
2423imp 410 . . . . . . 7 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ Tr 𝐵) → (𝐵𝐴𝐵𝐴))
2516, 24sylan2 595 . . . . . 6 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐵𝐴𝐵𝐴))
26 dfon2lem3 33143 . . . . . . . . 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 498 . . . . . . 7 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → Tr 𝐴)
29 psseq1 4015 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
30 treq 5142 . . . . . . . . . . 11 (𝑦 = 𝐴 → (Tr 𝑦 ↔ Tr 𝐴))
3129, 30anbi12d 633 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦𝐵 ∧ Tr 𝑦) ↔ (𝐴𝐵 ∧ Tr 𝐴)))
32 eleq1 2877 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
3331, 32imbi12d 348 . . . . . . . . 9 (𝑦 = 𝐴 → (((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) ↔ ((𝐴𝐵 ∧ Tr 𝐴) → 𝐴𝐵)))
341, 33spcv 3554 . . . . . . . 8 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → ((𝐴𝐵 ∧ Tr 𝐴) → 𝐴𝐵))
3534expcomd 420 . . . . . . 7 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → (Tr 𝐴 → (𝐴𝐵𝐴𝐵)))
3628, 35mpan9 510 . . . . . 6 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐴𝐵))
3725, 36orim12d 962 . . . . 5 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐵𝐴𝐴𝐵) → (𝐵𝐴𝐴𝐵)))
3813, 37syl5bi 245 . . . 4 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐴𝐵𝐵𝐴) → (𝐵𝐴𝐴𝐵)))
3912, 38syl5bir 246 . . 3 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (((𝐴𝐵𝐵𝐴) ∧ ¬ 𝐴 = 𝐵) → (𝐵𝐴𝐴𝐵)))
403, 39mpand 694 . 2 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
41 3orrot 1089 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (𝐴 = 𝐵𝐵𝐴𝐴𝐵))
42 3orass 1087 . . . 4 ((𝐴 = 𝐵𝐵𝐴𝐴𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐵𝐴𝐴𝐵)))
43 df-or 845 . . . 4 ((𝐴 = 𝐵 ∨ (𝐵𝐴𝐴𝐵)) ↔ (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
4442, 43bitri 278 . . 3 ((𝐴 = 𝐵𝐵𝐴𝐴𝐵) ↔ (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
4541, 44bitri 278 . 2 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (¬ 𝐴 = 𝐵 → (𝐵𝐴𝐴𝐵)))
4640, 45sylibr 237 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  wa 399  wo 844  w3o 1083  wal 1536   = wceq 1538  wcel 2111  wral 3106  Vcvv 3441  wss 3881  wpss 3882  Tr wtr 5136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-pw 4499  df-sn 4526  df-pr 4528  df-uni 4801  df-iun 4883  df-tr 5137  df-suc 6165
This theorem is referenced by:  dfon2lem6  33146  dfon2  33150
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