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Theorem infeq5i 9132
 Description: Half of infeq5 9133. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5i (ω ∈ V → ∃𝑥 𝑥 𝑥)

Proof of Theorem infeq5i
StepHypRef Expression
1 difexg 5197 . 2 (ω ∈ V → (ω ∖ {∅}) ∈ V)
2 0ex 5177 . . . . 5 ∅ ∈ V
32snid 4558 . . . 4 ∅ ∈ {∅}
4 disj4 4355 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5 disj3 4350 . . . . . 6 ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
64, 5bitr3i 280 . . . . 5 (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7 peano1 7600 . . . . . . 7 ∅ ∈ ω
8 eleq2 2840 . . . . . . 7 (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
97, 8mpbii 236 . . . . . 6 (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
109eldifbd 3871 . . . . 5 (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
116, 10sylbi 220 . . . 4 (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
123, 11mt4 116 . . 3 (ω ∖ {∅}) ⊊ ω
13 unidif0 5228 . . . . 5 (ω ∖ {∅}) = ω
14 limom 7594 . . . . . 6 Lim ω
15 limuni 6229 . . . . . 6 (Lim ω → ω = ω)
1614, 15ax-mp 5 . . . . 5 ω = ω
1713, 16eqtr4i 2784 . . . 4 (ω ∖ {∅}) = ω
1817psseq2i 3996 . . 3 ((ω ∖ {∅}) ⊊ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
1912, 18mpbir 234 . 2 (ω ∖ {∅}) ⊊ (ω ∖ {∅})
20 psseq1 3993 . . . 4 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ 𝑥))
21 unieq 4809 . . . . 5 (𝑥 = (ω ∖ {∅}) → 𝑥 = (ω ∖ {∅}))
2221psseq2d 3999 . . . 4 (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2320, 22bitrd 282 . . 3 (𝑥 = (ω ∖ {∅}) → (𝑥 𝑥 ↔ (ω ∖ {∅}) ⊊ (ω ∖ {∅})))
2423spcegv 3515 . 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 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3409   ∖ cdif 3855   ∩ cin 3857   ⊊ wpss 3859  ∅c0 4225  {csn 4522  ∪ cuni 4798  Lim wlim 6170  ωcom 7579 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-tr 5139  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-om 7580 This theorem is referenced by:  infeq5  9133  inf5  9141
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