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Theorem onsucuni2 6981
 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 2686 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 503 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 eloni 5692 . . . . 5 (suc 𝐵 ∈ On → Ord suc 𝐵)
4 ordsuc 6961 . . . . . . . 8 (Ord 𝐵 ↔ Ord suc 𝐵)
5 ordunisuc 6979 . . . . . . . 8 (Ord 𝐵 suc 𝐵 = 𝐵)
64, 5sylbir 225 . . . . . . 7 (Ord suc 𝐵 suc 𝐵 = 𝐵)
7 suceq 5749 . . . . . . 7 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
86, 7syl 17 . . . . . 6 (Ord suc 𝐵 → suc suc 𝐵 = suc 𝐵)
9 ordunisuc 6979 . . . . . 6 (Ord suc 𝐵 suc suc 𝐵 = suc 𝐵)
108, 9eqtr4d 2658 . . . . 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 4410 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
13 suceq 5749 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1412, 13syl 17 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
15 suceq 5749 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
1615unieqd 4412 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
1714, 16eqeq12d 2636 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
1811, 17syl5ibr 236 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
1918anabsi7 859 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
20 eloni 5692 . . . 4 (𝐴 ∈ On → Ord 𝐴)
21 ordunisuc 6979 . . . 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 2655 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∪ cuni 4402  Ord word 5681  Oncon0 5682  suc csuc 5684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-suc 5688 This theorem is referenced by:  rankxplim3  8688  rankxpsuc  8689
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