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Theorem ordsucuniel 7819
Description: Given an element 𝐴 of the union of an ordinal 𝐵, suc 𝐴 is an element of 𝐵 itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
ordsucuniel (Ord 𝐵 → (𝐴 𝐵 ↔ suc 𝐴𝐵))

Proof of Theorem ordsucuniel
StepHypRef Expression
1 orduni 7784 . . 3 (Ord 𝐵 → Ord 𝐵)
2 ordelord 6385 . . . 4 ((Ord 𝐵𝐴 𝐵) → Ord 𝐴)
32ex 412 . . 3 (Ord 𝐵 → (𝐴 𝐵 → Ord 𝐴))
41, 3syl 17 . 2 (Ord 𝐵 → (𝐴 𝐵 → Ord 𝐴))
5 ordelord 6385 . . . 4 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord suc 𝐴)
6 ordsuc 7808 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
75, 6sylibr 233 . . 3 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord 𝐴)
87ex 412 . 2 (Ord 𝐵 → (suc 𝐴𝐵 → Ord 𝐴))
9 ordsson 7777 . . . . . 6 (Ord 𝐵𝐵 ⊆ On)
10 ordunisssuc 6469 . . . . . 6 ((𝐵 ⊆ On ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
119, 10sylan 579 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
12 ordtri1 6396 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
131, 12sylan 579 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
14 ordtri1 6396 . . . . . 6 ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴𝐵))
156, 14sylan2b 593 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴𝐵))
1611, 13, 153bitr3d 309 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (¬ 𝐴 𝐵 ↔ ¬ suc 𝐴𝐵))
1716con4bid 317 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 𝐵 ↔ suc 𝐴𝐵))
1817ex 412 . 2 (Ord 𝐵 → (Ord 𝐴 → (𝐴 𝐵 ↔ suc 𝐴𝐵)))
194, 8, 18pm5.21ndd 379 1 (Ord 𝐵 → (𝐴 𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wcel 2099  wss 3944   cuni 4903  Ord word 6362  Oncon0 6363  suc csuc 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-suc 6369
This theorem is referenced by:  dfac12lem1  10152  dfac12lem2  10153  naddsuc2  42735
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