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Theorem ssonprc 7774
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 3047 . 2 (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2 ssorduni 7765 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
3 ordeleqon 7768 . . . . . . . 8 (Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On))
42, 3sylib 217 . . . . . . 7 (𝐴 ⊆ On → ( 𝐴 ∈ On ∨ 𝐴 = On))
54orcomd 869 . . . . . 6 (𝐴 ⊆ On → ( 𝐴 = On ∨ 𝐴 ∈ On))
65ord 862 . . . . 5 (𝐴 ⊆ On → (¬ 𝐴 = On → 𝐴 ∈ On))
7 uniexr 7749 . . . . 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 7764 . . . 4 ¬ On ∈ V
11 uniexg 7729 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
12 eleq1 2821 . . . . 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 282 1 (𝐴 ⊆ On → (𝐴 ∉ V ↔ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 845   = wceq 1541  wcel 2106  wnel 3046  Vcvv 3474  wss 3948   cuni 4908  Ord word 6363  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  inaprc  10830
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