Theorem ssonprc 7785

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 3071	. 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		ssorduni 7777	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		ordeleqon 7780	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
4	2, 3	sylib 221	. . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
5	4	orcomd 884	. . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On))
6	5	ord 877	. . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On))
7		uniexr 7761	. . . . 5 ⊢ (∪ 𝐴 ∈ On → 𝐴 ∈ V)
8	6, 7	syl6 36	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → 𝐴 ∈ V))
9	8	con1d 146	. . 3 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V → ∪ 𝐴 = On))
10		onprc 7776	. . . 4 ⊢ ¬ On ∈ V
11		uniexg 7738	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
12		eleq1 2857	. . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V))
13	11, 12	imbitrid 247	. . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V))
14	10, 13	mtoi 202	. . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V)
15	9, 14	impbid1 228	. 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On))
16	1, 15	bitrid 286	1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ∉ wnel 3070  Vcvv 3463   ⊆ wss 3913  ∪ cuni 4876  Ord word 6360  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365

This theorem is referenced by:  inaprc  10820