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Theorem dfon2lem4 31427
Description: Lemma for dfon2 31433. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)
Hypotheses
Ref Expression
dfon2lem4.1 𝐴 ∈ V
dfon2lem4.2 𝐵 ∈ V
Assertion
Ref Expression
dfon2lem4 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐵𝐴))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem dfon2lem4
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 inss1 3816 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
21sseli 3583 . . . . . . . 8 ((𝐴𝐵) ∈ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐴)
3 dfon2lem4.1 . . . . . . . . . . . 12 𝐴 ∈ V
4 dfon2lem3 31426 . . . . . . . . . . . 12 (𝐴 ∈ V → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))
53, 4ax-mp 5 . . . . . . . . . . 11 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧))
65simprd 479 . . . . . . . . . 10 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → ∀𝑧𝐴 ¬ 𝑧𝑧)
7 eleq1 2686 . . . . . . . . . . . . 13 (𝑧 = (𝐴𝐵) → (𝑧𝑧 ↔ (𝐴𝐵) ∈ 𝑧))
8 eleq2 2687 . . . . . . . . . . . . 13 (𝑧 = (𝐴𝐵) → ((𝐴𝐵) ∈ 𝑧 ↔ (𝐴𝐵) ∈ (𝐴𝐵)))
97, 8bitrd 268 . . . . . . . . . . . 12 (𝑧 = (𝐴𝐵) → (𝑧𝑧 ↔ (𝐴𝐵) ∈ (𝐴𝐵)))
109notbid 308 . . . . . . . . . . 11 (𝑧 = (𝐴𝐵) → (¬ 𝑧𝑧 ↔ ¬ (𝐴𝐵) ∈ (𝐴𝐵)))
1110rspccv 3295 . . . . . . . . . 10 (∀𝑧𝐴 ¬ 𝑧𝑧 → ((𝐴𝐵) ∈ 𝐴 → ¬ (𝐴𝐵) ∈ (𝐴𝐵)))
126, 11syl 17 . . . . . . . . 9 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → ((𝐴𝐵) ∈ 𝐴 → ¬ (𝐴𝐵) ∈ (𝐴𝐵)))
1312adantr 481 . . . . . . . 8 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐴𝐵) ∈ 𝐴 → ¬ (𝐴𝐵) ∈ (𝐴𝐵)))
142, 13syl5 34 . . . . . . 7 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐴𝐵) ∈ (𝐴𝐵) → ¬ (𝐴𝐵) ∈ (𝐴𝐵)))
1514pm2.01d 181 . . . . . 6 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ¬ (𝐴𝐵) ∈ (𝐴𝐵))
16 elin 3779 . . . . . 6 ((𝐴𝐵) ∈ (𝐴𝐵) ↔ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵))
1715, 16sylnib 318 . . . . 5 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ¬ ((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵))
185simpld 475 . . . . . . . 8 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → Tr 𝐴)
19 dfon2lem4.2 . . . . . . . . . 10 𝐵 ∈ V
20 dfon2lem3 31426 . . . . . . . . . 10 (𝐵 ∈ V → (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → (Tr 𝐵 ∧ ∀𝑧𝐵 ¬ 𝑧𝑧)))
2119, 20ax-mp 5 . . . . . . . . 9 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → (Tr 𝐵 ∧ ∀𝑧𝐵 ¬ 𝑧𝑧))
2221simpld 475 . . . . . . . 8 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → Tr 𝐵)
23 trin 4728 . . . . . . . 8 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
2418, 22, 23syl2an 494 . . . . . . 7 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → Tr (𝐴𝐵))
253inex1 4764 . . . . . . . . 9 (𝐴𝐵) ∈ V
26 psseq1 3677 . . . . . . . . . . 11 (𝑥 = (𝐴𝐵) → (𝑥𝐴 ↔ (𝐴𝐵) ⊊ 𝐴))
27 treq 4723 . . . . . . . . . . 11 (𝑥 = (𝐴𝐵) → (Tr 𝑥 ↔ Tr (𝐴𝐵)))
2826, 27anbi12d 746 . . . . . . . . . 10 (𝑥 = (𝐴𝐵) → ((𝑥𝐴 ∧ Tr 𝑥) ↔ ((𝐴𝐵) ⊊ 𝐴 ∧ Tr (𝐴𝐵))))
29 eleq1 2686 . . . . . . . . . 10 (𝑥 = (𝐴𝐵) → (𝑥𝐴 ↔ (𝐴𝐵) ∈ 𝐴))
3028, 29imbi12d 334 . . . . . . . . 9 (𝑥 = (𝐴𝐵) → (((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ↔ (((𝐴𝐵) ⊊ 𝐴 ∧ Tr (𝐴𝐵)) → (𝐴𝐵) ∈ 𝐴)))
3125, 30spcv 3288 . . . . . . . 8 (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (((𝐴𝐵) ⊊ 𝐴 ∧ Tr (𝐴𝐵)) → (𝐴𝐵) ∈ 𝐴))
3231adantr 481 . . . . . . 7 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (((𝐴𝐵) ⊊ 𝐴 ∧ Tr (𝐴𝐵)) → (𝐴𝐵) ∈ 𝐴))
3324, 32mpan2d 709 . . . . . 6 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐴𝐵) ⊊ 𝐴 → (𝐴𝐵) ∈ 𝐴))
34 psseq1 3677 . . . . . . . . . . 11 (𝑦 = (𝐴𝐵) → (𝑦𝐵 ↔ (𝐴𝐵) ⊊ 𝐵))
35 treq 4723 . . . . . . . . . . 11 (𝑦 = (𝐴𝐵) → (Tr 𝑦 ↔ Tr (𝐴𝐵)))
3634, 35anbi12d 746 . . . . . . . . . 10 (𝑦 = (𝐴𝐵) → ((𝑦𝐵 ∧ Tr 𝑦) ↔ ((𝐴𝐵) ⊊ 𝐵 ∧ Tr (𝐴𝐵))))
37 eleq1 2686 . . . . . . . . . 10 (𝑦 = (𝐴𝐵) → (𝑦𝐵 ↔ (𝐴𝐵) ∈ 𝐵))
3836, 37imbi12d 334 . . . . . . . . 9 (𝑦 = (𝐴𝐵) → (((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) ↔ (((𝐴𝐵) ⊊ 𝐵 ∧ Tr (𝐴𝐵)) → (𝐴𝐵) ∈ 𝐵)))
3925, 38spcv 3288 . . . . . . . 8 (∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵) → (((𝐴𝐵) ⊊ 𝐵 ∧ Tr (𝐴𝐵)) → (𝐴𝐵) ∈ 𝐵))
4039adantl 482 . . . . . . 7 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (((𝐴𝐵) ⊊ 𝐵 ∧ Tr (𝐴𝐵)) → (𝐴𝐵) ∈ 𝐵))
4124, 40mpan2d 709 . . . . . 6 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐴𝐵) ⊊ 𝐵 → (𝐴𝐵) ∈ 𝐵))
4233, 41anim12d 585 . . . . 5 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (((𝐴𝐵) ⊊ 𝐴 ∧ (𝐴𝐵) ⊊ 𝐵) → ((𝐴𝐵) ∈ 𝐴 ∧ (𝐴𝐵) ∈ 𝐵)))
4317, 42mtod 189 . . . 4 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ¬ ((𝐴𝐵) ⊊ 𝐴 ∧ (𝐴𝐵) ⊊ 𝐵))
44 ianor 509 . . . 4 (¬ ((𝐴𝐵) ⊊ 𝐴 ∧ (𝐴𝐵) ⊊ 𝐵) ↔ (¬ (𝐴𝐵) ⊊ 𝐴 ∨ ¬ (𝐴𝐵) ⊊ 𝐵))
4543, 44sylib 208 . . 3 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (¬ (𝐴𝐵) ⊊ 𝐴 ∨ ¬ (𝐴𝐵) ⊊ 𝐵))
46 sspss 3689 . . . . 5 ((𝐴𝐵) ⊆ 𝐴 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) = 𝐴))
471, 46mpbi 220 . . . 4 ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) = 𝐴)
48 inss2 3817 . . . . 5 (𝐴𝐵) ⊆ 𝐵
49 sspss 3689 . . . . 5 ((𝐴𝐵) ⊆ 𝐵 ↔ ((𝐴𝐵) ⊊ 𝐵 ∨ (𝐴𝐵) = 𝐵))
5048, 49mpbi 220 . . . 4 ((𝐴𝐵) ⊊ 𝐵 ∨ (𝐴𝐵) = 𝐵)
51 orel1 397 . . . . . 6 (¬ (𝐴𝐵) ⊊ 𝐴 → (((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) = 𝐴) → (𝐴𝐵) = 𝐴))
52 orc 400 . . . . . 6 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵))
5351, 52syl6 35 . . . . 5 (¬ (𝐴𝐵) ⊊ 𝐴 → (((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) = 𝐴) → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵)))
54 orel1 397 . . . . . 6 (¬ (𝐴𝐵) ⊊ 𝐵 → (((𝐴𝐵) ⊊ 𝐵 ∨ (𝐴𝐵) = 𝐵) → (𝐴𝐵) = 𝐵))
55 olc 399 . . . . . 6 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵))
5654, 55syl6 35 . . . . 5 (¬ (𝐴𝐵) ⊊ 𝐵 → (((𝐴𝐵) ⊊ 𝐵 ∨ (𝐴𝐵) = 𝐵) → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵)))
5753, 56jaoa 532 . . . 4 ((¬ (𝐴𝐵) ⊊ 𝐴 ∨ ¬ (𝐴𝐵) ⊊ 𝐵) → ((((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) = 𝐴) ∧ ((𝐴𝐵) ⊊ 𝐵 ∨ (𝐴𝐵) = 𝐵)) → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵)))
5847, 50, 57mp2ani 713 . . 3 ((¬ (𝐴𝐵) ⊊ 𝐴 ∨ ¬ (𝐴𝐵) ⊊ 𝐵) → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵))
5945, 58syl 17 . 2 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵))
60 df-ss 3573 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
61 sseqin2 3800 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
6260, 61orbi12i 543 . 2 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵))
6359, 62sylibr 224 1 ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cin 3558  wss 3559  wpss 3560  Tr wtr 4717
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 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408  df-iun 4492  df-tr 4718  df-suc 5693
This theorem is referenced by:  dfon2lem5  31428
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