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Theorem ordsucuniel 6974
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 6944 . . 3 (Ord 𝐵 → Ord 𝐵)
2 ordelord 5706 . . . 4 ((Ord 𝐵𝐴 𝐵) → Ord 𝐴)
32ex 450 . . 3 (Ord 𝐵 → (𝐴 𝐵 → Ord 𝐴))
41, 3syl 17 . 2 (Ord 𝐵 → (𝐴 𝐵 → Ord 𝐴))
5 ordelord 5706 . . . 4 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord suc 𝐴)
6 ordsuc 6964 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
75, 6sylibr 224 . . 3 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord 𝐴)
87ex 450 . 2 (Ord 𝐵 → (suc 𝐴𝐵 → Ord 𝐴))
9 ordsson 6939 . . . . . 6 (Ord 𝐵𝐵 ⊆ On)
10 ordunisssuc 5791 . . . . . 6 ((𝐵 ⊆ On ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
119, 10sylan 488 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
12 ordtri1 5717 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
131, 12sylan 488 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
14 ordtri1 5717 . . . . . 6 ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴𝐵))
156, 14sylan2b 492 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴𝐵))
1611, 13, 153bitr3d 298 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (¬ 𝐴 𝐵 ↔ ¬ suc 𝐴𝐵))
1716con4bid 307 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 𝐵 ↔ suc 𝐴𝐵))
1817ex 450 . 2 (Ord 𝐵 → (Ord 𝐴 → (𝐴 𝐵 ↔ suc 𝐴𝐵)))
194, 8, 18pm5.21ndd 369 1 (Ord 𝐵 → (𝐴 𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  wss 3556   cuni 4404  Ord word 5683  Oncon0 5684  suc csuc 5686
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
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 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-tr 4715  df-eprel 4987  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-ord 5687  df-on 5688  df-suc 5690
This theorem is referenced by:  dfac12lem1  8912  dfac12lem2  8913
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