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Theorem onsucuni2 7529
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 2877 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 482 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 eloni 6169 . . . . 5 (suc 𝐵 ∈ On → Ord suc 𝐵)
4 ordsuc 7509 . . . . . . . 8 (Ord 𝐵 ↔ Ord suc 𝐵)
5 ordunisuc 7527 . . . . . . . 8 (Ord 𝐵 suc 𝐵 = 𝐵)
64, 5sylbir 238 . . . . . . 7 (Ord suc 𝐵 suc 𝐵 = 𝐵)
7 suceq 6224 . . . . . . 7 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
86, 7syl 17 . . . . . 6 (Ord suc 𝐵 → suc suc 𝐵 = suc 𝐵)
9 ordunisuc 7527 . . . . . 6 (Ord suc 𝐵 suc suc 𝐵 = suc 𝐵)
108, 9eqtr4d 2836 . . . . 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 4811 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
13 suceq 6224 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1412, 13syl 17 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
15 suceq 6224 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1615unieqd 4814 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
1714, 16eqeq12d 2814 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
1811, 17syl5ibr 249 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
1918anabsi7 670 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
20 eloni 6169 . . . 4 (𝐴 ∈ On → Ord 𝐴)
21 ordunisuc 7527 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
2220, 21syl 17 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
2322adantr 484 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
2419, 23eqtrd 2833 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111   cuni 4800  Ord word 6158  Oncon0 6159  suc csuc 6161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-suc 6165
This theorem is referenced by:  rankxplim3  9294  rankxpsuc  9295
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