MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssonprc Structured version   Visualization version   GIF version

Theorem ssonprc 7796
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 3037 . 2 (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V)
2 ssorduni 7787 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
3 ordeleqon 7790 . . . . . . . 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 7771 . . . . 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 7786 . . . 4 ¬ On ∈ V
11 uniexg 7751 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
12 eleq1 2814 . . . . 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 1534  wcel 2099  wnel 3036  Vcvv 3462  wss 3947   cuni 4913  Ord word 6375  Oncon0 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380
This theorem is referenced by:  inaprc  10879
  Copyright terms: Public domain W3C validator