Theorem kelac2 43812

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 43811	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
4		cmptop 23552	. . 3 ⊢ ((topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
5	2, 3, 4	3syl 19	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
6		uncom 4112	. . . . . . 7 ⊢ (𝑆 ∪ {𝒫 ∪ 𝑆}) = ({𝒫 ∪ 𝑆} ∪ 𝑆)
7	6	difeq1i 4077	. . . . . 6 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆)
8		difun2 4442	. . . . . 6 ⊢ (({𝒫 ∪ 𝑆} ∪ 𝑆) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
9	7, 8	eqtri 2786	. . . . 5 ⊢ ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆) = ({𝒫 ∪ 𝑆} ∖ 𝑆)
10		snex 5410	. . . . . . 7 ⊢ {𝒫 ∪ 𝑆} ∈ V
11		uniprg 4888	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ {𝒫 ∪ 𝑆} ∈ V) → ∪ {𝑆, {𝒫 ∪ 𝑆}} = (𝑆 ∪ {𝒫 ∪ 𝑆}))
12	2, 10, 11	sylancl 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ {𝑆, {𝒫 ∪ 𝑆}} = (𝑆 ∪ {𝒫 ∪ 𝑆}))
13	12	difeq1d 4080	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = ((𝑆 ∪ {𝒫 ∪ 𝑆}) ∖ 𝑆))
14		incom 4162	. . . . . . 7 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
15		pwuninel 8267	. . . . . . . . 9 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
16	15	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
17		disjsn 4677	. . . . . . . 8 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
18	16, 17	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅)
19	14, 18	eqtrid 2810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅)
20		disj3 4414	. . . . . 6 ⊢ (({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅ ↔ {𝒫 ∪ 𝑆} = ({𝒫 ∪ 𝑆} ∖ 𝑆))
21	19, 20	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} = ({𝒫 ∪ 𝑆} ∖ 𝑆))
22	9, 13, 21	3eqtr4a 2824	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) = {𝒫 ∪ 𝑆})
23		prex 5409	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
24		bastg 23123	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ V → {𝑆, {𝒫 ∪ 𝑆}} ⊆ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
25	23, 24	mp1i 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝑆, {𝒫 ∪ 𝑆}} ⊆ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
26	10	prid2 4729	. . . . . 6 ⊢ {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}}
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ {𝑆, {𝒫 ∪ 𝑆}})
28	25, 27	sseldd 3938	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
29	22, 28	eqeltrd 2863	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))
30		prid1g 4726	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ {𝑆, {𝒫 ∪ 𝑆}})
31		elssuni 4904	. . . . 5 ⊢ (𝑆 ∈ {𝑆, {𝒫 ∪ 𝑆}} → 𝑆 ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}})
32	2, 30, 31	3syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}})
33		unitg 23124	. . . . . . 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 2772	. . . . 5 ⊢ ∪ {𝑆, {𝒫 ∪ 𝑆}} = ∪ (topGen‘{𝑆, {𝒫 ∪ 𝑆}})
36	35	iscld2 23185	. . . 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 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∈ (Clsd‘(topGen‘{𝑆, {𝒫 ∪ 𝑆}})) ↔ (∪ {𝑆, {𝒫 ∪ 𝑆}} ∖ 𝑆) ∈ (topGen‘{𝑆, {𝒫 ∪ 𝑆}})))
38	29, 37	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ (Clsd‘(topGen‘{𝑆, {𝒫 ∪ 𝑆}})))
39		f1oi 6859	. . 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 14	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝒫 ∪ 𝑆} ⊆ ∪ {𝑆, {𝒫 ∪ 𝑆}})
43		uniexg 7738	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V)
44		pwexg 5349	. . . . 5 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
45		snidg 4626	. . . . 5 ⊢ (𝒫 ∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ {𝒫 ∪ 𝑆})
46	2, 43, 44, 45	4syl 20	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ {𝒫 ∪ 𝑆})
47	42, 46	sseldd 3938	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ ∪ {𝑆, {𝒫 ∪ 𝑆}})
48	47, 34	eleqtrrdi 2874	. 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 43810	1 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  {cpr 4591  ∪ cuni 4872   ↦ cmpt 5192   I cid 5555   ↾ cres 5663  –1-1-onto→wf1o 6535  ‘cfv 6536  Xcixp 8891  topGenctg 17485  ∏_tcpt 17486  Topctop 23050  Clsdccld 23173  Compccmp 23543

This theorem was proved from 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-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem 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-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  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-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7859  df-1o 8449  df-2o 8450  df-ixp 8892  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9367  df-topgen 17491  df-pt 17492  df-top 23051  df-bases 23103  df-cld 23176  df-cmp 23544

This theorem is referenced by:  dfac21  43813