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Theorem ssonprc 7496
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 3121 . 2 (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2 ssorduni 7489 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
3 ordeleqon 7492 . . . . . . . 8 (Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On))
42, 3sylib 219 . . . . . . 7 (𝐴 ⊆ On → ( 𝐴 ∈ On ∨ 𝐴 = On))
54orcomd 865 . . . . . 6 (𝐴 ⊆ On → ( 𝐴 = On ∨ 𝐴 ∈ On))
65ord 858 . . . . 5 (𝐴 ⊆ On → (¬ 𝐴 = On → 𝐴 ∈ On))
7 uniexr 7474 . . . . 5 ( 𝐴 ∈ On → 𝐴 ∈ V)
86, 7syl6 35 . . . 4 (𝐴 ⊆ On → (¬ 𝐴 = On → 𝐴 ∈ V))
98con1d 147 . . 3 (𝐴 ⊆ On → (¬ 𝐴 ∈ V → 𝐴 = On))
10 onprc 7488 . . . 4 ¬ On ∈ V
11 uniexg 7456 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
12 eleq1 2897 . . . . 5 ( 𝐴 = On → ( 𝐴 ∈ V ↔ On ∈ V))
1311, 12syl5ib 245 . . . 4 ( 𝐴 = On → (𝐴 ∈ V → On ∈ V))
1410, 13mtoi 200 . . 3 ( 𝐴 = On → ¬ 𝐴 ∈ V)
159, 14impbid1 226 . 2 (𝐴 ⊆ On → (¬ 𝐴 ∈ V ↔ 𝐴 = On))
161, 15syl5bb 284 1 (𝐴 ⊆ On → (𝐴 ∉ V ↔ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wo 841   = wceq 1528  wcel 2105  wnel 3120  Vcvv 3492  wss 3933   cuni 4830  Ord word 6183  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  inaprc  10246
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