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Theorem infeq5i 8484
Description: Half of infeq5 8485. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5i (ω ∈ V → ∃𝑥 𝑥 𝑥)

Proof of Theorem infeq5i
StepHypRef Expression
1 difexg 4773 . 2 (ω ∈ V → (ω ∖ {∅}) ∈ V)
2 0ex 4755 . . . . 5 ∅ ∈ V
32snid 4184 . . . 4 ∅ ∈ {∅}
4 disj4 4002 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5 disj3 3998 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
64, 5bitr3i 266 . . . . 5 (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7 peano1 7039 . . . . . . 7 ∅ ∈ ω
8 eleq2 2687 . . . . . . 7 (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
97, 8mpbii 223 . . . . . 6 (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
109eldifbd 3572 . . . . 5 (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
116, 10sylbi 207 . . . 4 (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
123, 11mt4 115 . . 3 (ω ∖ {∅}) ⊊ ω
13 unidif0 4803 . . . . 5 (ω ∖ {∅}) = ω
14 limom 7034 . . . . . 6 Lim ω
15 limuni 5749 . . . . . 6 (Lim ω → ω = ω)
1614, 15ax-mp 5 . . . . 5 ω = ω
1713, 16eqtr4i 2646 . . . 4 (ω ∖ {∅}) = ω
1817psseq2i 3680 . . 3 ((ω ∖ {∅}) ⊊ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
1912, 18mpbir 221 . 2 (ω ∖ {∅}) ⊊ (ω ∖ {∅})
20 psseq1 3677 . . . 4 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ 𝑥))
21 unieq 4415 . . . . 5 (𝑥 = (ω ∖ {∅}) → 𝑥 = (ω ∖ {∅}))
2221psseq2d 3683 . . . 4 (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2320, 22bitrd 268 . . 3 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2423spcegv 3283 . 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 1480  wex 1701  wcel 1987  Vcvv 3189  cdif 3556  cin 3558  wpss 3560  c0 3896  {csn 4153   cuni 4407  Lim wlim 5688  ωcom 7019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-om 7020
This theorem is referenced by:  infeq5  8485  inf5  8493
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