Theorem infeq5i 9601

Description: Half of infeq5 9602. (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 5270	. . . . 5 ⊢ ∅ ∈ V
3	2	snid 4628	. . . 4 ⊢ ∅ ∈ {∅}
4		disj4 4419	. . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5		disj3 4414	. . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
6	4, 5	bitr3i 280	. . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7		peano1 7881	. . . . . . 7 ⊢ ∅ ∈ ω
8		eleq2 2852	. . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
9	7, 8	mpbii 236	. . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
10	9	eldifbd 3918	. . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
11	6, 10	sylbi 220	. . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
12	3, 11	mt4 117	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
13		unidif0 5330	. . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω
14		limom 7874	. . . . . 6 ⊢ Lim ω
15		limuni 6423	. . . . . 6 ⊢ (Lim ω → ω = ∪ ω)
16	14, 15	ax-mp 5	. . . . 5 ⊢ ω = ∪ ω
17	13, 16	eqtr4i 2789	. . . 4 ⊢ ∪ (ω ∖ {∅}) = ω
18	17	psseq2i 4047	. . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
19	12, 18	mpbir 234	. 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})
20		psseq1 4044	. . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥))
21		unieq 4883	. . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅}))
22	21	psseq2d 4050	. . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})))
23	20, 22	bitrd 282	. . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})))
24	23	spcegv 3556	. 2 ⊢ ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) → ∃𝑥 𝑥 ⊊ ∪ 𝑥))
25	1, 19, 24	mpisyl 22	1 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904   ⊊ wpss 3906  ∅c0 4286  {csn 4589  ∪ cuni 4872  Lim wlim 6361  ωcom 7858

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-om 7859

This theorem is used by:  infeq5  9602  inf5  9610