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Theorem infeq5i 9631
Description: Half of infeq5 9632. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5i (ω ∈ V → ∃𝑥 𝑥 𝑥)

Proof of Theorem infeq5i
StepHypRef Expression
1 difexg 5328 . 2 (ω ∈ V → (ω ∖ {∅}) ∈ V)
2 0ex 5308 . . . . 5 ∅ ∈ V
32snid 4665 . . . 4 ∅ ∈ {∅}
4 disj4 4459 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5 disj3 4454 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
64, 5bitr3i 277 . . . . 5 (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7 peano1 7879 . . . . . . 7 ∅ ∈ ω
8 eleq2 2823 . . . . . . 7 (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
97, 8mpbii 232 . . . . . 6 (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
109eldifbd 3962 . . . . 5 (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
116, 10sylbi 216 . . . 4 (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
123, 11mt4 116 . . 3 (ω ∖ {∅}) ⊊ ω
13 unidif0 5359 . . . . 5 (ω ∖ {∅}) = ω
14 limom 7871 . . . . . 6 Lim ω
15 limuni 6426 . . . . . 6 (Lim ω → ω = ω)
1614, 15ax-mp 5 . . . . 5 ω = ω
1713, 16eqtr4i 2764 . . . 4 (ω ∖ {∅}) = ω
1817psseq2i 4091 . . 3 ((ω ∖ {∅}) ⊊ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
1912, 18mpbir 230 . 2 (ω ∖ {∅}) ⊊ (ω ∖ {∅})
20 psseq1 4088 . . . 4 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ 𝑥))
21 unieq 4920 . . . . 5 (𝑥 = (ω ∖ {∅}) → 𝑥 = (ω ∖ {∅}))
2221psseq2d 4094 . . . 4 (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2320, 22bitrd 279 . . 3 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2423spcegv 3588 . 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 3946  cin 3948  wpss 3950  c0 4323  {csn 4629   cuni 4909  Lim wlim 6366  ωcom 7855
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7856
This theorem is referenced by:  infeq5  9632  inf5  9640
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