Theorem kelac2 43493

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 43492	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
4		cmptop 23360	. . 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 4098	. . . . . . 7 ⊢ (𝑆 ∪ {𝒫 ∪ 𝑆}) = ({𝒫 ∪ 𝑆} ∪ 𝑆)
7	6	difeq1i 4062	. . . . . 6 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆)
8		difun2 4421	. . . . . 6 ⊢ (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
9	7, 8	eqtri 2759	. . . . 5 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
10		snex 5381	. . . . . . 7 ⊢ {𝒫 ∪ 𝑆} ∈ V
11		uniprg 4866	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ {𝒫 ∪ 𝑆} ∈ V) → ∪ {𝑆, {𝒫 ∪ 𝑆}} = (𝑆 ∪ {𝒫 ∪ 𝑆}))
12	2, 10, 11	sylancl 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ {𝑆, {𝒫 ∪ 𝑆}} = (𝑆 ∪ {𝒫 ∪ 𝑆}))
13	12	difeq1d 4065	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆))
14		incom 4149	. . . . . . 7 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
15		pwuninel 8225	. . . . . . . . 9 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
16	15	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
17		disjsn 4655	. . . . . . . 8 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
18	16, 17	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅)
19	14, 18	eqtrid 2783	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅)
20		disj3 4394	. . . . . 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 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = {𝒫 ∪ 𝑆})
23		prex 5380	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
24		bastg 22931	. . . . . 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 4707	. . . . . 6 ⊢ {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}}
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}})
28	25, 27	sseldd 3922	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
29	22, 28	eqeltrd 2836	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
30		prid1g 4704	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ {𝑆, {𝒫 ∪ 𝑆}})
31		elssuni 4881	. . . . 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 22932	. . . . . . 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 2745	. . . . 5 ⊢ ∪ {𝑆, {𝒫 ∪ 𝑆}} = ∪ (topGen‘{𝑆, {𝒫 ∪ 𝑆}})
36	35	iscld2 22993	. . . 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 585	. . 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 6818	. . 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 4881	. . . . 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 7694	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
44		pwexg 5320	. . . . 5 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
45		snidg 4604	. . . . 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 3922	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ ∪ {𝑆, {𝒫 ∪ 𝑆}})
48	47, 34	eleqtrrdi 2847	. 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 43491	1 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567  {cpr 4569  ∪ cuni 4850   ↦ cmpt 5166   I cid 5525   ↾ cres 5633  –1-1-onto→wf1o 6497  ‘cfv 6498  Xcixp 8845  topGenctg 17400  ∏_tcpt 17401  Topctop 22858  Clsdccld 22981  Compccmp 23351

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-2o 8406  df-ixp 8846  df-en 8894  df-dom 8895  df-fin 8897  df-fi 9324  df-topgen 17406  df-pt 17407  df-top 22859  df-bases 22911  df-cld 22984  df-cmp 23352

This theorem is referenced by:  dfac21  43494