Theorem ordtoplem 31604

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 2782	. 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴)
2		ordeleqon 6880	. . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		unon 6923	. . . . . . . . 9 ⊢ ∪ On = On
4	3	eqcomi 2619	. . . . . . . 8 ⊢ On = ∪ On
5		id 22	. . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On)
6		unieq 4380	. . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
7	4, 5, 6	3eqtr4a 2670	. . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴)
8	7	orim2i 539	. . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
9	2, 8	sylbi 206	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴))
10	9	orcomd 402	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On))
11	10	ord 391	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On))
12		orduniorsuc 6922	. . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
13	12	ord 391	. . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
14		onuni 6885	. . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
15		ordtoplem.1	. . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)
16		eleq1a 2683	. . . 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 68	. 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ 𝑆))
19	1, 18	syl5bi 231	1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∪ cuni 4372  Ord word 5639  Oncon0 5640  suc csuc 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646

This theorem is referenced by:  ordtop  31605  ordtopcon  31608  ordtopt0  31611