Theorem ordtoplem 36578

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 2931	. 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴)
2		ordeleqon 7725	. . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		unon 7771	. . . . . . . . 9 ⊢ ∪ On = On
4	3	eqcomi 2743	. . . . . . . 8 ⊢ On = ∪ On
5		id 22	. . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On)
6		unieq 4872	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
7	4, 5, 6	3eqtr4a 2795	. . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴)
8	7	orim2i 910	. . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
9	2, 8	sylbi 217	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
10	9	orcomd 871	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On))
11	10	ord 864	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On))
12		orduniorsuc 7770	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
13	12	ord 864	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
14		onuni 7731	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
15		ordtoplem.1	. . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)
16		eleq1a 2829	. . . 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	biimtrid 242	1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∪ cuni 4861  Ord word 6314  Oncon0 6315  suc csuc 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319  df-suc 6321

This theorem is referenced by:  ordtop  36579  ordtopconn  36582  ordtopt0  36585