Theorem ordtoplem 37223

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 2956	. 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴)
2		ordeleqon 7796	. . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		unon 7842	. . . . . . . . 9 ⊢ ∪ On = On
4	3	eqcomi 2769	. . . . . . . 8 ⊢ On = ∪ On
5		id 23	. . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On)
6		unieq 4877	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
7	4, 5, 6	3eqtr4a 2821	. . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴)
8	7	orim2i 924	. . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
9	2, 8	sylbi 220	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
10	9	orcomd 885	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On))
11	10	ord 878	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On))
12		orduniorsuc 7841	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
13	12	ord 878	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
14		onuni 7802	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
15		ordtoplem.1	. . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)
16		eleq1a 2855	. . . 4 ⊢ (suc ∪ 𝐴 ∈ 𝑆 → (𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆))
17	14, 15, 16	3syl 19	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆))
18	11, 13, 17	syl6c 71	. 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ 𝑆))
19	1, 18	biimtrid 245	1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∪ cuni 4866  Ord word 6361  Oncon0 6362  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-suc 6368

This theorem is used by:  ordtop  37224  ordtopconn  37227  ordtopt0  37230