Theorem infeq5i 9605

Description: Half of infeq5 9606. (Contributed by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
infeq5i	⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)

Proof of Theorem infeq5i

Step	Hyp	Ref	Expression
1		difexg 5300	. 2 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V)
2		0ex 5272	. . . . 5 ⊢ ∅ ∈ V
3	2	snid 4633	. . . 4 ⊢ ∅ ∈ {∅}
4		disj4 4425	. . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5		disj3 4420	. . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
6	4, 5	bitr3i 280	. . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7		peano1 7885	. . . . . . 7 ⊢ ∅ ∈ ω
8		eleq2 2858	. . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
9	7, 8	mpbii 236	. . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
10	9	eldifbd 3926	. . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
11	6, 10	sylbi 220	. . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
12	3, 11	mt4 117	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
13		unidif0 5331	. . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω
14		limom 7878	. . . . . 6 ⊢ Lim ω
15		limuni 6424	. . . . . 6 ⊢ (Lim ω → ω = ∪ ω)
16	14, 15	ax-mp 5	. . . . 5 ⊢ ω = ∪ ω
17	13, 16	eqtr4i 2795	. . . 4 ⊢ ∪ (ω ∖ {∅}) = ω
18	17	psseq2i 4055	. . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
19	12, 18	mpbir 234	. 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})
20		psseq1 4052	. . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥))
21		unieq 4887	. . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅}))
22	21	psseq2d 4058	. . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})))
23	20, 22	bitrd 282	. . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})))
24	23	spcegv 3565	. 2 ⊢ ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) → ∃𝑥 𝑥 ⊊ ∪ 𝑥))
25	1, 19, 24	mpisyl 22	1 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   ⊊ wpss 3914  ∅c0 4294  {csn 4594  ∪ cuni 4876  Lim wlim 6362  ωcom 7862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7863

This theorem is referenced by:  infeq5  9606  inf5  9614