Theorem ssonprc 7770

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 3062	. 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		ssorduni 7762	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		ordeleqon 7765	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
4	2, 3	sylib 220	. . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
5	4	orcomd 882	. . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On))
6	5	ord 875	. . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On))
7		uniexr 7746	. . . . 5 ⊢ (∪ 𝐴 ∈ On → 𝐴 ∈ V)
8	6, 7	syl6 35	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → 𝐴 ∈ V))
9	8	con1d 145	. . 3 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V → ∪ 𝐴 = On))
10		onprc 7761	. . . 4 ⊢ ¬ On ∈ V
11		uniexg 7723	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
12		eleq1 2850	. . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V))
13	11, 12	imbitrid 246	. . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V))
14	10, 13	mtoi 201	. . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V)
15	9, 14	impbid1 227	. 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On))
16	1, 15	bitrid 285	1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ∉ wnel 3061  Vcvv 3454   ⊆ wss 3904  ∪ cuni 4865  Ord word 6345  Oncon0 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350

This theorem is referenced by:  inaprc  10794