Theorem ssonprc 6884

Description: Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)

Assertion

Ref	Expression
ssonprc	⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On))

Proof of Theorem ssonprc

Step	Hyp	Ref	Expression
1		df-nel 2783	. 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		ssorduni 6877	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		ordeleqon 6880	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
4	2, 3	sylib 207	. . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
5	4	orcomd 402	. . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On))
6	5	ord 391	. . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On))
7		elex 3185	. . . . . 6 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ∈ V)
8		uniexb 6866	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
9	7, 8	sylibr 223	. . . . 5 ⊢ (∪ 𝐴 ∈ On → 𝐴 ∈ V)
10	6, 9	syl6 34	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → 𝐴 ∈ V))
11	10	con1d 138	. . 3 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V → ∪ 𝐴 = On))
12		onprc 6876	. . . 4 ⊢ ¬ On ∈ V
13		uniexg 6853	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
14		eleq1 2676	. . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V))
15	13, 14	syl5ib 233	. . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V))
16	12, 15	mtoi 189	. . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V)
17	11, 16	impbid1 214	. 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On))
18	1, 17	syl5bb 271	1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372  Ord word 5639  Oncon0 5640

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-pow 4769  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-nel 2783  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-pw 4110  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

This theorem is referenced by:  inaprc  9537