Theorem ordtoplem 33778

Description: Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)

Hypothesis

Ref	Expression
ordtoplem.1	⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)

Assertion

Ref	Expression
ordtoplem	⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))

Proof of Theorem ordtoplem

Step	Hyp	Ref	Expression
1		df-ne 3017	. 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴)
2		ordeleqon 7497	. . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		unon 7540	. . . . . . . . 9 ⊢ ∪ On = On
4	3	eqcomi 2830	. . . . . . . 8 ⊢ On = ∪ On
5		id 22	. . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On)
6		unieq 4840	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
7	4, 5, 6	3eqtr4a 2882	. . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴)
8	7	orim2i 907	. . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
9	2, 8	sylbi 219	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
10	9	orcomd 867	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On))
11	10	ord 860	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On))
12		orduniorsuc 7539	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
13	12	ord 860	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
14		onuni 7502	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
15		ordtoplem.1	. . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)
16		eleq1a 2908	. . . 4 ⊢ (suc ∪ 𝐴 ∈ 𝑆 → (𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆))
17	14, 15, 16	3syl 18	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆))
18	11, 13, 17	syl6c 70	. 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ 𝑆))
19	1, 18	syl5bi 244	1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∪ cuni 4832  Ord word 6185  Oncon0 6186  suc csuc 6188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-tr 5166  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-ord 6189  df-on 6190  df-suc 6192

This theorem is referenced by:  ordtop  33779  ordtopconn  33782  ordtopt0  33785