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Theorem onsucuni2 7824
Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onsucuni2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)

Proof of Theorem onsucuni2
StepHypRef Expression
1 eleq1 2819 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 477 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 eloni 6373 . . . . 5 (suc 𝐵 ∈ On → Ord suc 𝐵)
4 ordsuc 7803 . . . . . . . 8 (Ord 𝐵 ↔ Ord suc 𝐵)
5 ordunisuc 7822 . . . . . . . 8 (Ord 𝐵 suc 𝐵 = 𝐵)
64, 5sylbir 234 . . . . . . 7 (Ord suc 𝐵 suc 𝐵 = 𝐵)
7 suceq 6429 . . . . . . 7 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
86, 7syl 17 . . . . . 6 (Ord suc 𝐵 → suc suc 𝐵 = suc 𝐵)
9 ordunisuc 7822 . . . . . 6 (Ord suc 𝐵 suc suc 𝐵 = suc 𝐵)
108, 9eqtr4d 2773 . . . . 5 (Ord suc 𝐵 → suc suc 𝐵 = suc suc 𝐵)
112, 3, 103syl 18 . . . 4 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc suc 𝐵 = suc suc 𝐵)
12 unieq 4918 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
13 suceq 6429 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1412, 13syl 17 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
15 suceq 6429 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1615unieqd 4921 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
1714, 16eqeq12d 2746 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
1811, 17imbitrrid 245 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
1918anabsi7 667 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
20 eloni 6373 . . . 4 (𝐴 ∈ On → Ord 𝐴)
21 ordunisuc 7822 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
2220, 21syl 17 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
2322adantr 479 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
2419, 23eqtrd 2770 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104   cuni 4907  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367  df-suc 6369
This theorem is referenced by:  rankxplim3  9878  rankxpsuc  9879  onsucf1lem  42321
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