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Theorem fin23lem24 9736
 Description: Lemma for fin23 9803. 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 766 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐴)
2 simplr 768 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐵𝐴)
3 simprl 770 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐵)
42, 3sseldd 3916 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐶𝐴)
5 ordelord 6182 . . . . . 6 ((Ord 𝐴𝐶𝐴) → Ord 𝐶)
61, 4, 5syl2anc 587 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐶)
7 simprr 772 . . . . . . 7 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐵)
82, 7sseldd 3916 . . . . . 6 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → 𝐷𝐴)
9 ordelord 6182 . . . . . 6 ((Ord 𝐴𝐷𝐴) → Ord 𝐷)
101, 8, 9syl2anc 587 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → Ord 𝐷)
11 ordtri3 6196 . . . . . 6 ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶𝐷𝐷𝐶)))
1211necon2abid 3029 . . . . 5 ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
136, 10, 12syl2anc 587 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) ↔ 𝐶𝐷))
14 simpr 488 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐷)
15 simplrl 776 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶𝐵)
1614, 15elind 4121 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → 𝐶 ∈ (𝐷𝐵))
176adantr 484 . . . . . . . . . 10 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → Ord 𝐶)
18 ordirr 6178 . . . . . . . . . 10 (Ord 𝐶 → ¬ 𝐶𝐶)
1917, 18syl 17 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶𝐶)
20 elinel1 4122 . . . . . . . . 9 (𝐶 ∈ (𝐶𝐵) → 𝐶𝐶)
2119, 20nsyl 142 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → ¬ 𝐶 ∈ (𝐶𝐵))
22 nelne1 3083 . . . . . . . 8 ((𝐶 ∈ (𝐷𝐵) ∧ ¬ 𝐶 ∈ (𝐶𝐵)) → (𝐷𝐵) ≠ (𝐶𝐵))
2316, 21, 22syl2anc 587 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐷𝐵) ≠ (𝐶𝐵))
2423necomd 3042 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐶𝐷) → (𝐶𝐵) ≠ (𝐷𝐵))
25 simpr 488 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐶)
26 simplrr 777 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷𝐵)
2725, 26elind 4121 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → 𝐷 ∈ (𝐶𝐵))
2810adantr 484 . . . . . . . . 9 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → Ord 𝐷)
29 ordirr 6178 . . . . . . . . 9 (Ord 𝐷 → ¬ 𝐷𝐷)
3028, 29syl 17 . . . . . . . 8 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷𝐷)
31 elinel1 4122 . . . . . . . 8 (𝐷 ∈ (𝐷𝐵) → 𝐷𝐷)
3230, 31nsyl 142 . . . . . . 7 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → ¬ 𝐷 ∈ (𝐷𝐵))
33 nelne1 3083 . . . . . . 7 ((𝐷 ∈ (𝐶𝐵) ∧ ¬ 𝐷 ∈ (𝐷𝐵)) → (𝐶𝐵) ≠ (𝐷𝐵))
3427, 32, 33syl2anc 587 . . . . . 6 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ 𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵))
3524, 34jaodan 955 . . . . 5 ((((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) ∧ (𝐶𝐷𝐷𝐶)) → (𝐶𝐵) ≠ (𝐷𝐵))
3635ex 416 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐷𝐷𝐶) → (𝐶𝐵) ≠ (𝐷𝐵)))
3713, 36sylbird 263 . . 3 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (𝐶𝐷 → (𝐶𝐵) ≠ (𝐷𝐵)))
3837necon4d 3011 . 2 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) → 𝐶 = 𝐷))
39 ineq1 4131 . 2 (𝐶 = 𝐷 → (𝐶𝐵) = (𝐷𝐵))
4038, 39impbid1 228 1 (((Ord 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) ↔ 𝐶 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∩ cin 3880   ⊆ wss 3881  Ord word 6159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-tr 5138  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-ord 6163 This theorem is referenced by:  fin23lem23  9740
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