Theorem kelac2 43061

Description: Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Hypotheses

Ref	Expression
kelac2.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉)
kelac2.z	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)
kelac2.k	⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))) ∈ Comp)

Assertion

Ref	Expression
kelac2	⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐼

Allowed substitution hints:   𝑆(𝑥)   𝑉(𝑥)

Proof of Theorem kelac2

Step	Hyp	Ref	Expression
1		kelac2.z	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)
2		kelac2.s	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉)
3		kelac2lem 43060	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
4		cmptop 23289	. . 3 ⊢ ((topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
5	2, 3, 4	3syl 18	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
6		uncom 4124	. . . . . . 7 ⊢ (𝑆 ∪ {𝒫 ∪ 𝑆}) = ({𝒫 ∪ 𝑆} ∪ 𝑆)
7	6	difeq1i 4088	. . . . . 6 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆)
8		difun2 4447	. . . . . 6 ⊢ (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
9	7, 8	eqtri 2753	. . . . 5 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
10		snex 5394	. . . . . . 7 ⊢ {𝒫 ∪ 𝑆} ∈ V
11		uniprg 4890	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ {𝒫 ∪ 𝑆} ∈ V) → ∪ {𝑆, {𝒫 ∪ 𝑆}} = (𝑆 ∪ {𝒫 ∪ 𝑆}))
12	2, 10, 11	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ {𝑆, {𝒫 ∪ 𝑆}} = (𝑆 ∪ {𝒫 ∪ 𝑆}))
13	12	difeq1d 4091	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆))
14		incom 4175	. . . . . . 7 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
15		pwuninel 8257	. . . . . . . . 9 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
16	15	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
17		disjsn 4678	. . . . . . . 8 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
18	16, 17	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅)
19	14, 18	eqtrid 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅)
20		disj3 4420	. . . . . 6 ⊢ (({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅ ↔ {𝒫 ∪ 𝑆} = ({𝒫 ∪ 𝑆} ∖ 𝑆))
21	19, 20	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} = ({𝒫 ∪ 𝑆} ∖ 𝑆))
22	9, 13, 21	3eqtr4a 2791	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = {𝒫 ∪ 𝑆})
23		prex 5395	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
24		bastg 22860	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ V → {𝑆, {𝒫 ∪ 𝑆}} ⊆ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
25	23, 24	mp1i 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝑆, {𝒫 ∪ 𝑆}} ⊆ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
26	10	prid2 4730	. . . . . 6 ⊢ {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}}
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}})
28	25, 27	sseldd 3950	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
29	22, 28	eqeltrd 2829	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
30		prid1g 4727	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ {𝑆, {𝒫 ∪ 𝑆}})
31		elssuni 4904	. . . . 5 ⊢ (𝑆 ∈ {𝑆, {𝒫 ∪ 𝑆}} → 𝑆 ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}})
32	2, 30, 31	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}})
33		unitg 22861	. . . . . . 7 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ V → ∪ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) = ∪ {𝑆, {𝒫 ∪ 𝑆}})
34	23, 33	ax-mp 5	. . . . . 6 ⊢ ∪ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) = ∪ {𝑆, {𝒫 ∪ 𝑆}}
35	34	eqcomi 2739	. . . . 5 ⊢ ∪ {𝑆, {𝒫 ∪ 𝑆}} = ∪ (topGen‘{𝑆, {𝒫 ∪ 𝑆}})
36	35	iscld2 22922	. . . 4 ⊢ (((topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top ∧ 𝑆 ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}}) → (𝑆 ∈ (Clsd‘(topGen‘{𝑆, {𝒫 ∪ 𝑆}})) ↔ (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}})))
37	5, 32, 36	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∈ (Clsd‘(topGen‘{𝑆, {𝒫 ∪ 𝑆}})) ↔ (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}})))
38	29, 37	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ (Clsd‘(topGen‘{𝑆, {𝒫 ∪ 𝑆}})))
39		f1oi 6841	. . 3 ⊢ ( I ↾ 𝑆):𝑆–1-1-onto→𝑆
40	39	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( I ↾ 𝑆):𝑆–1-1-onto→𝑆)
41		elssuni 4904	. . . . 5 ⊢ ({𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}} → {𝒫 ∪ 𝑆} ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}})
42	26, 41	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}})
43		uniexg 7719	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
44		pwexg 5336	. . . . 5 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
45		snidg 4627	. . . . 5 ⊢ (𝒫 ∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ {𝒫 ∪ 𝑆})
46	2, 43, 44, 45	4syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ {𝒫 ∪ 𝑆})
47	42, 46	sseldd 3950	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ ∪ {𝑆, {𝒫 ∪ 𝑆}})
48	47, 34	eleqtrrdi 2840	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ ∪ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
49		kelac2.k	. 2 ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))) ∈ Comp)
50	1, 5, 38, 40, 48, 49	kelac1 43059	1 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  {cpr 4594  ∪ cuni 4874   ↦ cmpt 5191   I cid 5535   ↾ cres 5643  –1-1-onto→wf1o 6513  ‘cfv 6514  Xcixp 8873  topGenctg 17407  ∏_tcpt 17408  Topctop 22787  Clsdccld 22910  Compccmp 23280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-om 7846  df-1o 8437  df-2o 8438  df-ixp 8874  df-en 8922  df-dom 8923  df-fin 8925  df-fi 9369  df-topgen 17413  df-pt 17414  df-top 22788  df-bases 22840  df-cld 22913  df-cmp 23281

This theorem is referenced by:  dfac21  43062