MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordsucuniel Structured version   Visualization version   GIF version

Theorem ordsucuniel 7603
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 7573 . . 3 (Ord 𝐵 → Ord 𝐵)
2 ordelord 6235 . . . 4 ((Ord 𝐵𝐴 𝐵) → Ord 𝐴)
32ex 416 . . 3 (Ord 𝐵 → (𝐴 𝐵 → Ord 𝐴))
41, 3syl 17 . 2 (Ord 𝐵 → (𝐴 𝐵 → Ord 𝐴))
5 ordelord 6235 . . . 4 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord suc 𝐴)
6 ordsuc 7593 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
75, 6sylibr 237 . . 3 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord 𝐴)
87ex 416 . 2 (Ord 𝐵 → (suc 𝐴𝐵 → Ord 𝐴))
9 ordsson 7567 . . . . . 6 (Ord 𝐵𝐵 ⊆ On)
10 ordunisssuc 6315 . . . . . 6 ((𝐵 ⊆ On ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
119, 10sylan 583 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
12 ordtri1 6246 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
131, 12sylan 583 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
14 ordtri1 6246 . . . . . 6 ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴𝐵))
156, 14sylan2b 597 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴𝐵))
1611, 13, 153bitr3d 312 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (¬ 𝐴 𝐵 ↔ ¬ suc 𝐴𝐵))
1716con4bid 320 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 𝐵 ↔ suc 𝐴𝐵))
1817ex 416 . 2 (Ord 𝐵 → (Ord 𝐴 → (𝐴 𝐵 ↔ suc 𝐴𝐵)))
194, 8, 18pm5.21ndd 384 1 (Ord 𝐵 → (𝐴 𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2110  wss 3866   cuni 4819  Ord word 6212  Oncon0 6213  suc csuc 6215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217  df-suc 6219
This theorem is referenced by:  dfac12lem1  9757  dfac12lem2  9758
  Copyright terms: Public domain W3C validator