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Theorem ssonprc 7727
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
StepHypRef Expression
1 df-nel 3051 . 2 (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2 ssorduni 7718 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
3 ordeleqon 7721 . . . . . . . 8 (Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On))
42, 3sylib 217 . . . . . . 7 (𝐴 ⊆ On → ( 𝐴 ∈ On ∨ 𝐴 = On))
54orcomd 870 . . . . . 6 (𝐴 ⊆ On → ( 𝐴 = On ∨ 𝐴 ∈ On))
65ord 863 . . . . 5 (𝐴 ⊆ On → (¬ 𝐴 = On → 𝐴 ∈ On))
7 uniexr 7702 . . . . 5 ( 𝐴 ∈ On → 𝐴 ∈ V)
86, 7syl6 35 . . . 4 (𝐴 ⊆ On → (¬ 𝐴 = On → 𝐴 ∈ V))
98con1d 145 . . 3 (𝐴 ⊆ On → (¬ 𝐴 ∈ V → 𝐴 = On))
10 onprc 7717 . . . 4 ¬ On ∈ V
11 uniexg 7682 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
12 eleq1 2826 . . . . 5 ( 𝐴 = On → ( 𝐴 ∈ V ↔ On ∈ V))
1311, 12imbitrid 243 . . . 4 ( 𝐴 = On → (𝐴 ∈ V → On ∈ V))
1410, 13mtoi 198 . . 3 ( 𝐴 = On → ¬ 𝐴 ∈ V)
159, 14impbid1 224 . 2 (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ 𝐴 = On))
161, 15bitrid 283 1 (𝐴 ⊆ On → (𝐴 ∉ V ↔ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 846   = wceq 1542  wcel 2107  wnel 3050  Vcvv 3448  wss 3915   cuni 4870  Ord word 6321  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  inaprc  10779
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