Theorem ordtoplem 33896

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 2988	. 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴)
2		ordeleqon 7483	. . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		unon 7526	. . . . . . . . 9 ⊢ ∪ On = On
4	3	eqcomi 2807	. . . . . . . 8 ⊢ On = ∪ On
5		id 22	. . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On)
6		unieq 4811	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
7	4, 5, 6	3eqtr4a 2859	. . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴)
8	7	orim2i 908	. . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
9	2, 8	sylbi 220	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
10	9	orcomd 868	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On))
11	10	ord 861	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On))
12		orduniorsuc 7525	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
13	12	ord 861	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
14		onuni 7488	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
15		ordtoplem.1	. . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)
16		eleq1a 2885	. . . 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 245	1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∪ 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:  ordtop  33897  ordtopconn  33900  ordtopt0  33903