Theorem kelac2 43042

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 43041	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
4		cmptop 23280	. . 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 4109	. . . . . . 7 ⊢ (𝑆 ∪ {𝒫 ∪ 𝑆}) = ({𝒫 ∪ 𝑆} ∪ 𝑆)
7	6	difeq1i 4073	. . . . . 6 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆)
8		difun2 4432	. . . . . 6 ⊢ (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
9	7, 8	eqtri 2752	. . . . 5 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
10		snex 5375	. . . . . . 7 ⊢ {𝒫 ∪ 𝑆} ∈ V
11		uniprg 4874	. . . . . . 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 4076	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆))
14		incom 4160	. . . . . . 7 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
15		pwuninel 8208	. . . . . . . . 9 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
16	15	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
17		disjsn 4663	. . . . . . . 8 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
18	16, 17	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅)
19	14, 18	eqtrid 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅)
20		disj3 4405	. . . . . 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 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = {𝒫 ∪ 𝑆})
23		prex 5376	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
24		bastg 22851	. . . . . 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 4715	. . . . . 6 ⊢ {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}}
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}})
28	25, 27	sseldd 3936	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
29	22, 28	eqeltrd 2828	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
30		prid1g 4712	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ {𝑆, {𝒫 ∪ 𝑆}})
31		elssuni 4888	. . . . 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 22852	. . . . . . 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 2738	. . . . 5 ⊢ ∪ {𝑆, {𝒫 ∪ 𝑆}} = ∪ (topGen‘{𝑆, {𝒫 ∪ 𝑆}})
36	35	iscld2 22913	. . . 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 6802	. . 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 4888	. . . . 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 7676	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
44		pwexg 5317	. . . . 5 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
45		snidg 4612	. . . . 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 3936	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ ∪ {𝑆, {𝒫 ∪ 𝑆}})
48	47, 34	eleqtrrdi 2839	. 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 43040	1 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  {cpr 4579  ∪ cuni 4858   ↦ cmpt 5173   I cid 5513   ↾ cres 5621  –1-1-onto→wf1o 6481  ‘cfv 6482  Xcixp 8824  topGenctg 17341  ∏_tcpt 17342  Topctop 22778  Clsdccld 22901  Compccmp 23271

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-2o 8389  df-ixp 8825  df-en 8873  df-dom 8874  df-fin 8876  df-fi 9301  df-topgen 17347  df-pt 17348  df-top 22779  df-bases 22831  df-cld 22904  df-cmp 23272

This theorem is referenced by:  dfac21  43043