Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.7 Structured version   Visualization version   GIF version

Theorem tz7.7 5666
 Description: A transitive class belongs to an ordinal class iff it is strictly included in it. Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.)
Assertion
Ref Expression
tz7.7 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴)))

Proof of Theorem tz7.7
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtr 5654 . . . 4 (Ord 𝐴 → Tr 𝐴)
2 ordfr 5655 . . . 4 (Ord 𝐴 → E Fr 𝐴)
3 tz7.2 5022 . . . . 5 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
433exp 1256 . . . 4 (Tr 𝐴 → ( E Fr 𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴))))
51, 2, 4sylc 63 . . 3 (Ord 𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
65adantr 480 . 2 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
7 pssdifn0 3898 . . . . . 6 ((𝐵𝐴𝐵𝐴) → (𝐴𝐵) ≠ ∅)
8 difss 3699 . . . . . . . . . . . 12 (𝐴𝐵) ⊆ 𝐴
9 tz7.5 5661 . . . . . . . . . . . 12 ((Ord 𝐴 ∧ (𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅)
108, 9mp3an2 1404 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅)
11 eldifi 3694 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
12 trss 4689 . . . . . . . . . . . . . . . . . 18 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
13 difin0ss 3900 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝐵) ∩ 𝑥) = ∅ → (𝑥𝐴𝑥𝐵))
1413com12 32 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵))
1511, 12, 14syl56 35 . . . . . . . . . . . . . . . . 17 (Tr 𝐴 → (𝑥 ∈ (𝐴𝐵) → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵)))
161, 15syl 17 . . . . . . . . . . . . . . . 16 (Ord 𝐴 → (𝑥 ∈ (𝐴𝐵) → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵)))
1716ad2antrr 758 . . . . . . . . . . . . . . 15 (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) → (𝑥 ∈ (𝐴𝐵) → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵)))
1817imp32 448 . . . . . . . . . . . . . 14 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥𝐵)
19 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
2019biimpcd 238 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦𝐵 → (𝑦 = 𝑥𝑥𝐵))
21 eldifn 3695 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
2220, 21nsyli 154 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑦 = 𝑥))
2322imp 444 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐵𝑥 ∈ (𝐴𝐵)) → ¬ 𝑦 = 𝑥)
2423adantll 746 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵)) → ¬ 𝑦 = 𝑥)
2524adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑦 = 𝑥)
26 trel 4687 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Tr 𝐵 → ((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
2726expcomd 453 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Tr 𝐵 → (𝑦𝐵 → (𝑥𝑦𝑥𝐵)))
2827imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Tr 𝐵𝑦𝐵) → (𝑥𝑦𝑥𝐵))
2928, 21nsyli 154 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Tr 𝐵𝑦𝐵) → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝑦))
3029ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (Tr 𝐵 → (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝑦)))
3130adantld 482 . . . . . . . . . . . . . . . . . . . . . 22 (Tr 𝐵 → ((𝐵𝐴𝑦𝐵) → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝑦)))
3231imp32 448 . . . . . . . . . . . . . . . . . . . . 21 ((Tr 𝐵 ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑥𝑦)
3332adantll 746 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑥𝑦)
34 ordwe 5653 . . . . . . . . . . . . . . . . . . . . . 22 (Ord 𝐴 → E We 𝐴)
35 ssel2 3563 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵𝐴𝑦𝐵) → 𝑦𝐴)
3635, 11anim12i 588 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝑦𝐴𝑥𝐴))
37 wecmpep 5030 . . . . . . . . . . . . . . . . . . . . . 22 (( E We 𝐴 ∧ (𝑦𝐴𝑥𝐴)) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
3834, 36, 37syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((Ord 𝐴 ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
3938adantlr 747 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
4025, 33, 39ecase23d 1428 . . . . . . . . . . . . . . . . . . 19 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → 𝑦𝑥)
4140exp44 639 . . . . . . . . . . . . . . . . . 18 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → 𝑦𝑥))))
4241com34 89 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝑥 ∈ (𝐴𝐵) → (𝑦𝐵𝑦𝑥))))
4342imp31 447 . . . . . . . . . . . . . . . 16 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝑦𝐵𝑦𝑥))
4443ssrdv 3574 . . . . . . . . . . . . . . 15 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → 𝐵𝑥)
4544adantrr 749 . . . . . . . . . . . . . 14 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝐵𝑥)
4618, 45eqssd 3585 . . . . . . . . . . . . 13 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥 = 𝐵)
4711ad2antrl 760 . . . . . . . . . . . . 13 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥𝐴)
4846, 47eqeltrrd 2689 . . . . . . . . . . . 12 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝐵𝐴)
4948rexlimdvaa 3014 . . . . . . . . . . 11 (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) → (∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅ → 𝐵𝐴))
5010, 49syl5 33 . . . . . . . . . 10 (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) → ((Ord 𝐴 ∧ (𝐴𝐵) ≠ ∅) → 𝐵𝐴))
5150exp4b 630 . . . . . . . . 9 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (Ord 𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴))))
5251com23 84 . . . . . . . 8 ((Ord 𝐴 ∧ Tr 𝐵) → (Ord 𝐴 → (𝐵𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴))))
5352adantrd 483 . . . . . . 7 ((Ord 𝐴 ∧ Tr 𝐵) → ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴))))
5453pm2.43i 50 . . . . . 6 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴)))
557, 54syl7 72 . . . . 5 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → ((𝐵𝐴𝐵𝐴) → 𝐵𝐴)))
5655exp4a 631 . . . 4 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴))))
5756pm2.43d 51 . . 3 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
5857impd 446 . 2 ((Ord 𝐴 ∧ Tr 𝐵) → ((𝐵𝐴𝐵𝐴) → 𝐵𝐴))
596, 58impbid 201 1 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  Tr wtr 4680   E cep 4947   Fr wfr 4994   We wwe 4996  Ord word 5639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643 This theorem is referenced by:  ordelssne  5667  dfon2  30941
 Copyright terms: Public domain W3C validator