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Theorem infeq5i 9633
Description: Half of infeq5 9634. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5i (ω ∈ V → ∃𝑥 𝑥 𝑥)

Proof of Theorem infeq5i
StepHypRef Expression
1 difexg 5327 . 2 (ω ∈ V → (ω ∖ {∅}) ∈ V)
2 0ex 5307 . . . . 5 ∅ ∈ V
32snid 4664 . . . 4 ∅ ∈ {∅}
4 disj4 4458 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5 disj3 4453 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
64, 5bitr3i 276 . . . . 5 (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7 peano1 7881 . . . . . . 7 ∅ ∈ ω
8 eleq2 2822 . . . . . . 7 (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
97, 8mpbii 232 . . . . . 6 (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
109eldifbd 3961 . . . . 5 (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
116, 10sylbi 216 . . . 4 (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
123, 11mt4 116 . . 3 (ω ∖ {∅}) ⊊ ω
13 unidif0 5358 . . . . 5 (ω ∖ {∅}) = ω
14 limom 7873 . . . . . 6 Lim ω
15 limuni 6425 . . . . . 6 (Lim ω → ω = ω)
1614, 15ax-mp 5 . . . . 5 ω = ω
1713, 16eqtr4i 2763 . . . 4 (ω ∖ {∅}) = ω
1817psseq2i 4090 . . 3 ((ω ∖ {∅}) ⊊ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
1912, 18mpbir 230 . 2 (ω ∖ {∅}) ⊊ (ω ∖ {∅})
20 psseq1 4087 . . . 4 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ 𝑥))
21 unieq 4919 . . . . 5 (𝑥 = (ω ∖ {∅}) → 𝑥 = (ω ∖ {∅}))
2221psseq2d 4093 . . . 4 (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2320, 22bitrd 278 . . 3 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2423spcegv 3587 . 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 1541  wex 1781  wcel 2106  Vcvv 3474  cdif 3945  cin 3947  wpss 3949  c0 4322  {csn 4628   cuni 4908  Lim wlim 6365  ωcom 7857
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-pr 5427  ax-un 7727
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-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  df-lim 6369  df-suc 6370  df-om 7858
This theorem is referenced by:  infeq5  9634  inf5  9642
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