Theorem ssonprc 7109

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 3000	. 2 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2		ssorduni 7102	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		ordeleqon 7105	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
4	2, 3	sylib 208	. . . . . . 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		uniexr 7089	. . . . 5 ⊢ (∪ 𝐴 ∈ On → 𝐴 ∈ V)
8	6, 7	syl6 35	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 = On → 𝐴 ∈ V))
9	8	con1d 139	. . 3 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V → ∪ 𝐴 = On))
10		onprc 7101	. . . 4 ⊢ ¬ On ∈ V
11		uniexg 7072	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
12		eleq1 2791	. . . . 5 ⊢ (∪ 𝐴 = On → (∪ 𝐴 ∈ V ↔ On ∈ V))
13	11, 12	syl5ib 234	. . . 4 ⊢ (∪ 𝐴 = On → (𝐴 ∈ V → On ∈ V))
14	10, 13	mtoi 190	. . 3 ⊢ (∪ 𝐴 = On → ¬ 𝐴 ∈ V)
15	9, 14	impbid1 215	. 2 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ ∪ 𝐴 = On))
16	1, 15	syl5bb 272	1 ⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   = wceq 1596   ∈ wcel 2103   ∉ wnel 2999  Vcvv 3304   ⊆ wss 3680  ∪ cuni 4544  Ord word 5835  Oncon0 5836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-tr 4861  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-ord 5839  df-on 5840

This theorem is referenced by:  inaprc  9771