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Theorem fin23lem24 10319
Description: Lemma for fin23 10386. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
fin23lem24 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) ↔ 𝐶 = 𝐷))

Proof of Theorem fin23lem24
StepHypRef Expression
1 simpll 763 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐴)
2 simplr 765 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐵𝐴)
3 simprl 767 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐵)
42, 3sseldd 3982 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐴)
5 ordelord 6385 . . . . . 6 ((Ord 𝐴𝐶𝐴) → Ord 𝐶)
61, 4, 5syl2anc 582 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐶)
7 simprr 769 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐵)
82, 7sseldd 3982 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐴)
9 ordelord 6385 . . . . . 6 ((Ord 𝐴𝐷𝐴) → Ord 𝐷)
101, 8, 9syl2anc 582 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐷)
11 ordtri3 6399 . . . . . 6 ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶𝐷𝐷𝐶)))
1211necon2abid 2981 . . . . 5 ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
136, 10, 12syl2anc 582 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
14 simpr 483 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐷)
15 simplrl 773 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐵)
1614, 15elind 4193 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶 ∈ (𝐷𝐵))
176adantr 479 . . . . . . . . . 10 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → Ord 𝐶)
18 ordirr 6381 . . . . . . . . . 10 (Ord 𝐶 → ¬ 𝐶𝐶)
1917, 18syl 17 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶𝐶)
20 elinel1 4194 . . . . . . . . 9 (𝐶 ∈ (𝐶𝐵) → 𝐶𝐶)
2119, 20nsyl 140 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶 ∈ (𝐶𝐵))
22 nelne1 3037 . . . . . . . 8 ((𝐶 ∈ (𝐷𝐵) ∧ ¬ 𝐶 ∈ (𝐶𝐵)) → (𝐷𝐵) ≠ (𝐶𝐵))
2316, 21, 22syl2anc 582 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐷𝐵) ≠ (𝐶𝐵))
2423necomd 2994 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐶𝐵) ≠ (𝐷𝐵))
25 simpr 483 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐶)
26 simplrr 774 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐵)
2725, 26elind 4193 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷 ∈ (𝐶𝐵))
2810adantr 479 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → Ord 𝐷)
29 ordirr 6381 . . . . . . . . 9 (Ord 𝐷 → ¬ 𝐷𝐷)
3028, 29syl 17 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷𝐷)
31 elinel1 4194 . . . . . . . 8 (𝐷 ∈ (𝐷𝐵) → 𝐷𝐷)
3230, 31nsyl 140 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷 ∈ (𝐷𝐵))
33 nelne1 3037 . . . . . . 7 ((𝐷 ∈ (𝐶𝐵) ∧ ¬ 𝐷 ∈ (𝐷𝐵)) → (𝐶𝐵) ≠ (𝐷𝐵))
3427, 32, 33syl2anc 582 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵))
3524, 34jaodan 954 . . . . 5 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ (𝐶𝐷𝐷𝐶)) → (𝐶𝐵) ≠ (𝐷𝐵))
3635ex 411 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵)))
3713, 36sylbird 259 . . 3 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (𝐶𝐷 → (𝐶𝐵) ≠ (𝐷𝐵)))
3837necon4d 2962 . 2 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) → 𝐶 = 𝐷))
39 ineq1 4204 . 2 (𝐶 = 𝐷 → (𝐶𝐵) = (𝐷𝐵))
4038, 39impbid1 224 1 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) ↔ 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 843   = wceq 1539  wcel 2104  wne 2938  cin 3946  wss 3947  Ord word 6362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366
This theorem is referenced by:  fin23lem23  10323
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