Theorem ssonprc 7730

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 3039	. 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		ssorduni 7722	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		ordeleqon 7725	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
4	2, 3	sylib 219	. . . . . . 7 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
5	4	orcomd 877	. . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On ∨ ∪ 𝐴 ∈ On))
6	5	ord 870	. . . . 5 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → ∪ 𝐴 ∈ On))
7		uniexr 7706	. . . . 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 7721	. . . 4 ⊢ ¬ On ∈ V
11		uniexg 7683	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
12		eleq1 2827	. . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V))
13	11, 12	imbitrid 245	. . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V))
14	10, 13	mtoi 200	. . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V)
15	9, 14	impbid1 226	. 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On))
16	1, 15	bitrid 284	1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ∉ wnel 3038  Vcvv 3431   ⊆ wss 3883  ∪ cuni 4838  Ord word 6309  Oncon0 6310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314

This theorem is referenced by:  inaprc  10750