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Theorem onsucuni2 7538
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 2897 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 479 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 eloni 6194 . . . . 5 (suc 𝐵 ∈ On → Ord suc 𝐵)
4 ordsuc 7518 . . . . . . . 8 (Ord 𝐵 ↔ Ord suc 𝐵)
5 ordunisuc 7536 . . . . . . . 8 (Ord 𝐵 suc 𝐵 = 𝐵)
64, 5sylbir 236 . . . . . . 7 (Ord suc 𝐵 suc 𝐵 = 𝐵)
7 suceq 6249 . . . . . . 7 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
86, 7syl 17 . . . . . 6 (Ord suc 𝐵 → suc suc 𝐵 = suc 𝐵)
9 ordunisuc 7536 . . . . . 6 (Ord suc 𝐵 suc suc 𝐵 = suc 𝐵)
108, 9eqtr4d 2856 . . . . 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 4838 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
13 suceq 6249 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1412, 13syl 17 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
15 suceq 6249 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1615unieqd 4840 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
1714, 16eqeq12d 2834 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
1811, 17syl5ibr 247 . . 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 6194 . . . 4 (𝐴 ∈ On → Ord 𝐴)
21 ordunisuc 7536 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
2220, 21syl 17 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
2322adantr 481 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
2419, 23eqtrd 2853 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   cuni 4830  Ord word 6183  Oncon0 6184  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190
This theorem is referenced by:  rankxplim3  9298  rankxpsuc  9299
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