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Theorem tz7.7 5787
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 5775 . . . 4 (Ord 𝐴 → Tr 𝐴)
2 ordfr 5776 . . . 4 (Ord 𝐴 → E Fr 𝐴)
3 tz7.2 5127 . . . . 5 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
433exp 1283 . . . 4 (Tr 𝐴 → ( E Fr 𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴))))
51, 2, 4sylc 65 . . 3 (Ord 𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
65adantr 480 . 2 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
7 pssdifn0 3977 . . . . . 6 ((𝐵𝐴𝐵𝐴) → (𝐴𝐵) ≠ ∅)
8 difss 3770 . . . . . . . . . . . 12 (𝐴𝐵) ⊆ 𝐴
9 tz7.5 5782 . . . . . . . . . . . 12 ((Ord 𝐴 ∧ (𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅)
108, 9mp3an2 1452 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅)
11 eldifi 3765 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
12 trss 4794 . . . . . . . . . . . . . . . . . 18 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
13 difin0ss 3979 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝐵) ∩ 𝑥) = ∅ → (𝑥𝐴𝑥𝐵))
1413com12 32 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵))
1511, 12, 14syl56 36 . . . . . . . . . . . . . . . . 17 (Tr 𝐴 → (𝑥 ∈ (𝐴𝐵) → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵)))
161, 15syl 17 . . . . . . . . . . . . . . . 16 (Ord 𝐴 → (𝑥 ∈ (𝐴𝐵) → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵)))
1716ad2antrr 762 . . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
2019biimpcd 239 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦𝐵 → (𝑦 = 𝑥𝑥𝐵))
21 eldifn 3766 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
2220, 21nsyli 155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑦 = 𝑥))
2322imp 444 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐵𝑥 ∈ (𝐴𝐵)) → ¬ 𝑦 = 𝑥)
2423adantll 750 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵)) → ¬ 𝑦 = 𝑥)
2524adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑦 = 𝑥)
26 trel 4792 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Tr 𝐵 → ((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
2726expcomd 453 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Tr 𝐵 → (𝑦𝐵 → (𝑥𝑦𝑥𝐵)))
2827imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Tr 𝐵𝑦𝐵) → (𝑥𝑦𝑥𝐵))
2928, 21nsyli 155 . . . . . . . . . . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑥𝑦)
34 ordwe 5774 . . . . . . . . . . . . . . . . . . . . . 22 (Ord 𝐴 → E We 𝐴)
35 ssel2 3631 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵𝐴𝑦𝐵) → 𝑦𝐴)
3635, 11anim12i 589 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝑦𝐴𝑥𝐴))
37 wecmpep 5135 . . . . . . . . . . . . . . . . . . . . . 22 (( E We 𝐴 ∧ (𝑦𝐴𝑥𝐴)) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
3834, 36, 37syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((Ord 𝐴 ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
3938adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
4025, 33, 39ecase23d 1476 . . . . . . . . . . . . . . . . . . 19 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → 𝑦𝑥)
4140exp44 640 . . . . . . . . . . . . . . . . . 18 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → 𝑦𝑥))))
4241com34 91 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝑥 ∈ (𝐴𝐵) → (𝑦𝐵𝑦𝑥))))
4342imp31 447 . . . . . . . . . . . . . . . 16 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝑦𝐵𝑦𝑥))
4443ssrdv 3642 . . . . . . . . . . . . . . 15 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → 𝐵𝑥)
4544adantrr 753 . . . . . . . . . . . . . 14 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝐵𝑥)
4618, 45eqssd 3653 . . . . . . . . . . . . 13 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥 = 𝐵)
4711ad2antrl 764 . . . . . . . . . . . . 13 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥𝐴)
4846, 47eqeltrrd 2731 . . . . . . . . . . . 12 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝐵𝐴)
4948rexlimdvaa 3061 . . . . . . . . . . 11 (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) → (∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅ → 𝐵𝐴))
5010, 49syl5 34 . . . . . . . . . 10 (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) → ((Ord 𝐴 ∧ (𝐴𝐵) ≠ ∅) → 𝐵𝐴))
5150exp4b 631 . . . . . . . . 9 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (Ord 𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴))))
5251com23 86 . . . . . . . 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 52 . . . . . 6 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴)))
557, 54syl7 74 . . . . 5 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → ((𝐵𝐴𝐵𝐴) → 𝐵𝐴)))
5655exp4a 632 . . . 4 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴))))
5756pm2.43d 53 . . 3 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
5857impd 446 . 2 ((Ord 𝐴 ∧ Tr 𝐵) → ((𝐵𝐴𝐵𝐴) → 𝐵𝐴))
596, 58impbid 202 1 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1053   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cdif 3604  cin 3606  wss 3607  c0 3948  Tr wtr 4785   E cep 5057   Fr wfr 5099   We wwe 5101  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by:  ordelssne  5788  dfon2  31821
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