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Theorem fin23lem24 9307
Description: Lemma for fin23 9374. 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 807 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐴)
2 simplr 809 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐵𝐴)
3 simprl 811 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐵)
42, 3sseldd 3733 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐴)
5 ordelord 5894 . . . . . 6 ((Ord 𝐴𝐶𝐴) → Ord 𝐶)
61, 4, 5syl2anc 696 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐶)
7 simprr 813 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐵)
82, 7sseldd 3733 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐴)
9 ordelord 5894 . . . . . 6 ((Ord 𝐴𝐷𝐴) → Ord 𝐷)
101, 8, 9syl2anc 696 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐷)
11 ordtri3 5908 . . . . . 6 ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶𝐷𝐷𝐶)))
1211necon2abid 2962 . . . . 5 ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
136, 10, 12syl2anc 696 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
14 simpr 479 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐷)
15 simplrl 819 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐵)
1614, 15elind 3929 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶 ∈ (𝐷𝐵))
176adantr 472 . . . . . . . . . 10 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → Ord 𝐶)
18 ordirr 5890 . . . . . . . . . 10 (Ord 𝐶 → ¬ 𝐶𝐶)
1917, 18syl 17 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶𝐶)
20 inss1 3964 . . . . . . . . . 10 (𝐶𝐵) ⊆ 𝐶
2120sseli 3728 . . . . . . . . 9 (𝐶 ∈ (𝐶𝐵) → 𝐶𝐶)
2219, 21nsyl 135 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶 ∈ (𝐶𝐵))
23 nelne1 3016 . . . . . . . 8 ((𝐶 ∈ (𝐷𝐵) ∧ ¬ 𝐶 ∈ (𝐶𝐵)) → (𝐷𝐵) ≠ (𝐶𝐵))
2416, 22, 23syl2anc 696 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐷𝐵) ≠ (𝐶𝐵))
2524necomd 2975 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐶𝐵) ≠ (𝐷𝐵))
26 simpr 479 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐶)
27 simplrr 820 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐵)
2826, 27elind 3929 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷 ∈ (𝐶𝐵))
2910adantr 472 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → Ord 𝐷)
30 ordirr 5890 . . . . . . . . 9 (Ord 𝐷 → ¬ 𝐷𝐷)
3129, 30syl 17 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷𝐷)
32 inss1 3964 . . . . . . . . 9 (𝐷𝐵) ⊆ 𝐷
3332sseli 3728 . . . . . . . 8 (𝐷 ∈ (𝐷𝐵) → 𝐷𝐷)
3431, 33nsyl 135 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷 ∈ (𝐷𝐵))
35 nelne1 3016 . . . . . . 7 ((𝐷 ∈ (𝐶𝐵) ∧ ¬ 𝐷 ∈ (𝐷𝐵)) → (𝐶𝐵) ≠ (𝐷𝐵))
3628, 34, 35syl2anc 696 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵))
3725, 36jaodan 861 . . . . 5 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ (𝐶𝐷𝐷𝐶)) → (𝐶𝐵) ≠ (𝐷𝐵))
3837ex 449 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵)))
3913, 38sylbird 250 . . 3 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (𝐶𝐷 → (𝐶𝐵) ≠ (𝐷𝐵)))
4039necon4d 2944 . 2 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) → 𝐶 = 𝐷))
41 ineq1 3938 . 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 382  wa 383   = wceq 1620  wcel 2127  wne 2920  cin 3702  wss 3703  Ord word 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-tr 4893  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-ord 5875
This theorem is referenced by:  fin23lem23  9311
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