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

Theorem ssonprc 7807
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 7799 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
3 ordeleqon 7802 . . . . . . . 8 (Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On))
42, 3sylib 218 . . . . . . 7 (𝐴 ⊆ On → ( 𝐴 ∈ On ∨ 𝐴 = On))
54orcomd 872 . . . . . 6 (𝐴 ⊆ On → ( 𝐴 = On ∨ 𝐴 ∈ On))
65ord 865 . . . . 5 (𝐴 ⊆ On → (¬ 𝐴 = On → 𝐴 ∈ On))
7 uniexr 7783 . . . . 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 7798 . . . 4 ¬ On ∈ V
11 uniexg 7760 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
12 eleq1 2829 . . . . 5 ( 𝐴 = On → ( 𝐴 ∈ V ↔ On ∈ V))
1311, 12imbitrid 244 . . . 4 ( 𝐴 = On → (𝐴 ∈ V → On ∈ V))
1410, 13mtoi 199 . . 3 ( 𝐴 = On → ¬ 𝐴 ∈ V)
159, 14impbid1 225 . 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 206  wo 848   = wceq 1540  wcel 2108  wnel 3046  Vcvv 3480  wss 3951   cuni 4907  Ord word 6383  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  inaprc  10876
  Copyright terms: Public domain W3C validator