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Theorem fin23lem24 9088
Description: Lemma for fin23 9155. 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 789 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐴)
2 simplr 791 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐵𝐴)
3 simprl 793 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐵)
42, 3sseldd 3584 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐴)
5 ordelord 5704 . . . . . 6 ((Ord 𝐴𝐶𝐴) → Ord 𝐶)
61, 4, 5syl2anc 692 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐶)
7 simprr 795 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐵)
82, 7sseldd 3584 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐴)
9 ordelord 5704 . . . . . 6 ((Ord 𝐴𝐷𝐴) → Ord 𝐷)
101, 8, 9syl2anc 692 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐷)
11 ordtri3 5718 . . . . . 6 ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶𝐷𝐷𝐶)))
1211necon2abid 2832 . . . . 5 ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
136, 10, 12syl2anc 692 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
14 simpr 477 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐷)
15 simplrl 799 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐵)
1614, 15elind 3776 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶 ∈ (𝐷𝐵))
176adantr 481 . . . . . . . . . 10 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → Ord 𝐶)
18 ordirr 5700 . . . . . . . . . 10 (Ord 𝐶 → ¬ 𝐶𝐶)
1917, 18syl 17 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶𝐶)
20 inss1 3811 . . . . . . . . . 10 (𝐶𝐵) ⊆ 𝐶
2120sseli 3579 . . . . . . . . 9 (𝐶 ∈ (𝐶𝐵) → 𝐶𝐶)
2219, 21nsyl 135 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶 ∈ (𝐶𝐵))
23 nelne1 2886 . . . . . . . 8 ((𝐶 ∈ (𝐷𝐵) ∧ ¬ 𝐶 ∈ (𝐶𝐵)) → (𝐷𝐵) ≠ (𝐶𝐵))
2416, 22, 23syl2anc 692 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐷𝐵) ≠ (𝐶𝐵))
2524necomd 2845 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐶𝐵) ≠ (𝐷𝐵))
26 simpr 477 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐶)
27 simplrr 800 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐵)
2826, 27elind 3776 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷 ∈ (𝐶𝐵))
2910adantr 481 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → Ord 𝐷)
30 ordirr 5700 . . . . . . . . 9 (Ord 𝐷 → ¬ 𝐷𝐷)
3129, 30syl 17 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷𝐷)
32 inss1 3811 . . . . . . . . 9 (𝐷𝐵) ⊆ 𝐷
3332sseli 3579 . . . . . . . 8 (𝐷 ∈ (𝐷𝐵) → 𝐷𝐷)
3431, 33nsyl 135 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷 ∈ (𝐷𝐵))
35 nelne1 2886 . . . . . . 7 ((𝐷 ∈ (𝐶𝐵) ∧ ¬ 𝐷 ∈ (𝐷𝐵)) → (𝐶𝐵) ≠ (𝐷𝐵))
3628, 34, 35syl2anc 692 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵))
3725, 36jaodan 825 . . . . 5 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ (𝐶𝐷𝐷𝐶)) → (𝐶𝐵) ≠ (𝐷𝐵))
3837ex 450 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵)))
3913, 38sylbird 250 . . 3 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (𝐶𝐷 → (𝐶𝐵) ≠ (𝐷𝐵)))
4039necon4d 2814 . 2 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) → 𝐶 = 𝐷))
41 ineq1 3785 . 2 (𝐶 = 𝐷 → (𝐶𝐵) = (𝐷𝐵))
4240, 41impbid1 215 1 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) ↔ 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  cin 3554  wss 3555  Ord word 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685
This theorem is referenced by:  fin23lem23  9092
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