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Theorem tz7.7 6210
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 6198 . . . 4 (Ord 𝐴 → Tr 𝐴)
2 ordfr 6199 . . . 4 (Ord 𝐴 → E Fr 𝐴)
3 tz7.2 5532 . . . . 5 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
433exp 1111 . . . 4 (Tr 𝐴 → ( E Fr 𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴))))
51, 2, 4sylc 65 . . 3 (Ord 𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
65adantr 481 . 2 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
7 pssdifn0 4322 . . . . . 6 ((𝐵𝐴𝐵𝐴) → (𝐴𝐵) ≠ ∅)
8 difss 4105 . . . . . . . . . . . 12 (𝐴𝐵) ⊆ 𝐴
9 tz7.5 6205 . . . . . . . . . . . 12 ((Ord 𝐴 ∧ (𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅)
108, 9mp3an2 1440 . . . . . . . . . . 11 ((Ord 𝐴 ∧ (𝐴𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴𝐵)((𝐴𝐵) ∩ 𝑥) = ∅)
11 eldifi 4100 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
12 trss 5172 . . . . . . . . . . . . . . . . . 18 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
13 difin0ss 4325 . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . 15 (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) → (𝑥 ∈ (𝐴𝐵) → (((𝐴𝐵) ∩ 𝑥) = ∅ → 𝑥𝐵)))
1817imp32 419 . . . . . . . . . . . . . 14 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥𝐵)
19 eleq1w 2892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
2019biimpcd 250 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦𝐵 → (𝑦 = 𝑥𝑥𝐵))
21 eldifn 4101 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
2220, 21nsyli 160 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑦 = 𝑥))
2322imp 407 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐵𝑥 ∈ (𝐴𝐵)) → ¬ 𝑦 = 𝑥)
2423adantll 710 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵)) → ¬ 𝑦 = 𝑥)
2524adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑦 = 𝑥)
26 trel 5170 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Tr 𝐵 → ((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
2726expcomd 417 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Tr 𝐵 → (𝑦𝐵 → (𝑥𝑦𝑥𝐵)))
2827imp 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Tr 𝐵𝑦𝐵) → (𝑥𝑦𝑥𝐵))
2928, 21nsyli 160 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Tr 𝐵𝑦𝐵) → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝑦))
3029ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (Tr 𝐵 → (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝑦)))
3130adantld 491 . . . . . . . . . . . . . . . . . . . . . 22 (Tr 𝐵 → ((𝐵𝐴𝑦𝐵) → (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝑦)))
3231imp32 419 . . . . . . . . . . . . . . . . . . . . 21 ((Tr 𝐵 ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑥𝑦)
3332adantll 710 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → ¬ 𝑥𝑦)
34 ordwe 6197 . . . . . . . . . . . . . . . . . . . . . 22 (Ord 𝐴 → E We 𝐴)
35 ssel2 3959 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵𝐴𝑦𝐵) → 𝑦𝐴)
3635, 11anim12i 612 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝑦𝐴𝑥𝐴))
37 wecmpep 5540 . . . . . . . . . . . . . . . . . . . . . 22 (( E We 𝐴 ∧ (𝑦𝐴𝑥𝐴)) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
3834, 36, 37syl2an 595 . . . . . . . . . . . . . . . . . . . . 21 ((Ord 𝐴 ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
3938adantlr 711 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → (𝑦𝑥𝑦 = 𝑥𝑥𝑦))
4025, 33, 39ecase23d 1464 . . . . . . . . . . . . . . . . . . 19 (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵𝐴𝑦𝐵) ∧ 𝑥 ∈ (𝐴𝐵))) → 𝑦𝑥)
4140exp44 438 . . . . . . . . . . . . . . . . . 18 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝑦𝐵 → (𝑥 ∈ (𝐴𝐵) → 𝑦𝑥))))
4241com34 91 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝑥 ∈ (𝐴𝐵) → (𝑦𝐵𝑦𝑥))))
4342imp31 418 . . . . . . . . . . . . . . . 16 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝑦𝐵𝑦𝑥))
4443ssrdv 3970 . . . . . . . . . . . . . . 15 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → 𝐵𝑥)
4544adantrr 713 . . . . . . . . . . . . . 14 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝐵𝑥)
4618, 45eqssd 3981 . . . . . . . . . . . . 13 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥 = 𝐵)
4711ad2antrl 724 . . . . . . . . . . . . 13 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝑥𝐴)
4846, 47eqeltrrd 2911 . . . . . . . . . . . 12 ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵𝐴) ∧ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐴𝐵) ∩ 𝑥) = ∅)) → 𝐵𝐴)
4948rexlimdvaa 3282 . . . . . . . . . . 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 431 . . . . . . . . 9 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (Ord 𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴))))
5251com23 86 . . . . . . . 8 ((Ord 𝐴 ∧ Tr 𝐵) → (Ord 𝐴 → (𝐵𝐴 → ((𝐴𝐵) ≠ ∅ → 𝐵𝐴))))
5352adantrd 492 . . . . . . 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 432 . . . 4 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴 → (𝐵𝐴𝐵𝐴))))
5756pm2.43d 53 . . 3 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 → (𝐵𝐴𝐵𝐴)))
5857impd 411 . 2 ((Ord 𝐴 ∧ Tr 𝐵) → ((𝐵𝐴𝐵𝐴) → 𝐵𝐴))
596, 58impbid 213 1 ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3o 1078   = wceq 1528  wcel 2105  wne 3013  wrex 3136  cdif 3930  cin 3932  wss 3933  c0 4288  Tr wtr 5163   E cep 5457   Fr wfr 5504   We wwe 5506  Ord word 6183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187
This theorem is referenced by:  ordelssne  6211  dfon2  32934
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