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Theorem ordsucuniel 7844
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 7809 . . 3 (Ord 𝐵 → Ord 𝐵)
2 ordelord 6406 . . . 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 6406 . . . 4 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord suc 𝐴)
6 ordsuc 7833 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
75, 6sylibr 234 . . 3 ((Ord 𝐵 ∧ suc 𝐴𝐵) → Ord 𝐴)
87ex 412 . 2 (Ord 𝐵 → (suc 𝐴𝐵 → Ord 𝐴))
9 ordsson 7803 . . . . . 6 (Ord 𝐵𝐵 ⊆ On)
10 ordunisssuc 6490 . . . . . 6 ((𝐵 ⊆ On ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
119, 10sylan 580 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴𝐵 ⊆ suc 𝐴))
12 ordtri1 6417 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
131, 12sylan 580 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ¬ 𝐴 𝐵))
14 ordtri1 6417 . . . . . 6 ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴𝐵))
156, 14sylan2b 594 . . . . 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 206  wa 395  wcel 2108  wss 3951   cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  naddsuc2  8739  dfac12lem1  10184  dfac12lem2  10185
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