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Theorem infeq5i 9618
Description: Half of infeq5 9619. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5i (ω ∈ V → ∃𝑥 𝑥 𝑥)

Proof of Theorem infeq5i
StepHypRef Expression
1 difexg 5323 . 2 (ω ∈ V → (ω ∖ {∅}) ∈ V)
2 0ex 5303 . . . . 5 ∅ ∈ V
32snid 4660 . . . 4 ∅ ∈ {∅}
4 disj4 4456 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5 disj3 4451 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
64, 5bitr3i 277 . . . . 5 (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7 peano1 7866 . . . . . . 7 ∅ ∈ ω
8 eleq2 2823 . . . . . . 7 (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
97, 8mpbii 232 . . . . . 6 (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
109eldifbd 3959 . . . . 5 (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
116, 10sylbi 216 . . . 4 (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
123, 11mt4 116 . . 3 (ω ∖ {∅}) ⊊ ω
13 unidif0 5354 . . . . 5 (ω ∖ {∅}) = ω
14 limom 7858 . . . . . 6 Lim ω
15 limuni 6417 . . . . . 6 (Lim ω → ω = ω)
1614, 15ax-mp 5 . . . . 5 ω = ω
1713, 16eqtr4i 2764 . . . 4 (ω ∖ {∅}) = ω
1817psseq2i 4088 . . 3 ((ω ∖ {∅}) ⊊ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
1912, 18mpbir 230 . 2 (ω ∖ {∅}) ⊊ (ω ∖ {∅})
20 psseq1 4085 . . . 4 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ 𝑥))
21 unieq 4915 . . . . 5 (𝑥 = (ω ∖ {∅}) → 𝑥 = (ω ∖ {∅}))
2221psseq2d 4091 . . . 4 (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2320, 22bitrd 279 . . 3 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2423spcegv 3586 . 2 ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊊ (ω ∖ {∅}) → ∃𝑥 𝑥 𝑥))
251, 19, 24mpisyl 21 1 (ω ∈ V → ∃𝑥 𝑥 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cdif 3943  cin 3945  wpss 3947  c0 4320  {csn 4624   cuni 4904  Lim wlim 6357  ωcom 7842
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 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712
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 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-tr 5262  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-om 7843
This theorem is referenced by:  infeq5  9619  inf5  9627
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