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Theorem fin23lem24 9027
Description: Lemma for fin23 9094. 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 786 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐴)
2 simplr 788 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐵𝐴)
3 simprl 790 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐵)
42, 3sseldd 3569 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐴)
5 ordelord 5662 . . . . . 6 ((Ord 𝐴𝐶𝐴) → Ord 𝐶)
61, 4, 5syl2anc 691 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐶)
7 simprr 792 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐵)
82, 7sseldd 3569 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐴)
9 ordelord 5662 . . . . . 6 ((Ord 𝐴𝐷𝐴) → Ord 𝐷)
101, 8, 9syl2anc 691 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐷)
11 ordtri3 5676 . . . . . 6 ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶𝐷𝐷𝐶)))
1211necon2abid 2824 . . . . 5 ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
136, 10, 12syl2anc 691 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
14 simpr 476 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐷)
15 simplrl 796 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐵)
1614, 15elind 3760 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶 ∈ (𝐷𝐵))
176adantr 480 . . . . . . . . . 10 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → Ord 𝐶)
18 ordirr 5658 . . . . . . . . . 10 (Ord 𝐶 → ¬ 𝐶𝐶)
1917, 18syl 17 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶𝐶)
20 inss1 3795 . . . . . . . . . 10 (𝐶𝐵) ⊆ 𝐶
2120sseli 3564 . . . . . . . . 9 (𝐶 ∈ (𝐶𝐵) → 𝐶𝐶)
2219, 21nsyl 134 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶 ∈ (𝐶𝐵))
23 nelne1 2878 . . . . . . . 8 ((𝐶 ∈ (𝐷𝐵) ∧ ¬ 𝐶 ∈ (𝐶𝐵)) → (𝐷𝐵) ≠ (𝐶𝐵))
2416, 22, 23syl2anc 691 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐷𝐵) ≠ (𝐶𝐵))
2524necomd 2837 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐶𝐵) ≠ (𝐷𝐵))
26 simpr 476 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐶)
27 simplrr 797 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐵)
2826, 27elind 3760 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷 ∈ (𝐶𝐵))
2910adantr 480 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → Ord 𝐷)
30 ordirr 5658 . . . . . . . . 9 (Ord 𝐷 → ¬ 𝐷𝐷)
3129, 30syl 17 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷𝐷)
32 inss1 3795 . . . . . . . . 9 (𝐷𝐵) ⊆ 𝐷
3332sseli 3564 . . . . . . . 8 (𝐷 ∈ (𝐷𝐵) → 𝐷𝐷)
3431, 33nsyl 134 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷 ∈ (𝐷𝐵))
35 nelne1 2878 . . . . . . 7 ((𝐷 ∈ (𝐶𝐵) ∧ ¬ 𝐷 ∈ (𝐷𝐵)) → (𝐶𝐵) ≠ (𝐷𝐵))
3628, 34, 35syl2anc 691 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵))
3725, 36jaodan 822 . . . . 5 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ (𝐶𝐷𝐷𝐶)) → (𝐶𝐵) ≠ (𝐷𝐵))
3837ex 449 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵)))
3913, 38sylbird 249 . . 3 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (𝐶𝐷 → (𝐶𝐵) ≠ (𝐷𝐵)))
4039necon4d 2806 . 2 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) → 𝐶 = 𝐷))
41 ineq1 3769 . 2 (𝐶 = 𝐷 → (𝐶𝐵) = (𝐷𝐵))
4240, 41impbid1 214 1 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) ↔ 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wne 2780  cin 3539  wss 3540  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-pss 3556  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:  fin23lem23  9031
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