Theorem infeq5i 9551

Description: Half of infeq5 9552. (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 5267	. 2 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V)
2		0ex 5243	. . . . 5 ⊢ ∅ ∈ V
3	2	snid 4607	. . . 4 ⊢ ∅ ∈ {∅}
4		disj4 4400	. . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω)
5		disj3 4395	. . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
6	4, 5	bitr3i 277	. . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅}))
7		peano1 7834	. . . . . . 7 ⊢ ∅ ∈ ω
8		eleq2 2826	. . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
9	7, 8	mpbii 233	. . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
10	9	eldifbd 3903	. . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
11	6, 10	sylbi 217	. . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅})
12	3, 11	mt4 116	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
13		unidif0 5298	. . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω
14		limom 7827	. . . . . 6 ⊢ Lim ω
15		limuni 6380	. . . . . 6 ⊢ (Lim ω → ω = ∪ ω)
16	14, 15	ax-mp 5	. . . . 5 ⊢ ω = ∪ ω
17	13, 16	eqtr4i 2763	. . . 4 ⊢ ∪ (ω ∖ {∅}) = ω
18	17	psseq2i 4034	. . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω)
19	12, 18	mpbir 231	. 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})
20		psseq1 4031	. . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥))
21		unieq 4862	. . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅}))
22	21	psseq2d 4037	. . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})))
23	20, 22	bitrd 279	. . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅})))
24	23	spcegv 3540	. 2 ⊢ ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) → ∃𝑥 𝑥 ⊊ ∪ 𝑥))
25	1, 19, 24	mpisyl 21	1 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887   ∩ cin 3889   ⊊ wpss 3891  ∅c0 4274  {csn 4568  ∪ cuni 4851  Lim wlim 6319  ωcom 7811

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-om 7812

This theorem is referenced by:  infeq5  9552  inf5  9560