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Theorem ordsucuniel 7806
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 7771 . . 3 (Ord 𝐵 → Ord 𝐵)
2 ordelord 6377 . . . 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 6377 . . . 4 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord suc 𝐴)
6 ordsuc 7795 . . . 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 7764 . . . . . 6 (Ord 𝐵𝐵 ⊆ On)
10 ordunisssuc 6461 . . . . . 6 ((𝐵 ⊆ On ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
119, 10sylan 579 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
12 ordtri1 6388 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
131, 12sylan 579 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
14 ordtri1 6388 . . . . . 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 2098  wss 3941   cuni 4900  Ord word 6354  Oncon0 6355  suc csuc 6357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-suc 6361
This theorem is referenced by:  dfac12lem1  10135  dfac12lem2  10136  naddsuc2  42657
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