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Theorem phialllem2 4617

Description: Lemma for phiall 4618. Any set without 0c is equal to the Phi of a set. (Contributed by Scott Fenton, 8-Apr-2021.)

Hypothesis

Ref	Expression
phiall.1

Assertion

Ref	Expression
phialllem2	0c Phi

Distinct variable group:   ,

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Proof of Theorem phialllem2 Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3476	. . 3 Nn Nn
2		inss1 3475	. . . . 5 Nn
3	2	sseli 3269	. . . 4 0c Nn 0c
4	3	con3i 127	. . 3 0c 0c Nn
5		phiall.1	. . . . 5
6		nncex 4396	. . . . 5 Nn
7	5, 6	inex 4105	. . . 4 Nn
8	7	phialllem1 4616	. . 3 Nn Nn 0c Nn Nn Phi
9	1, 4, 8	sylancr 644	. 2 0c Nn Phi
10		uncom 3408	. . . . . . 7 Nn Nn Nn Nn
11		inundif 3628	. . . . . . 7 Nn Nn
12	10, 11	eqtri 2373	. . . . . 6 Nn Nn
13		uneq2 3412	. . . . . 6 Nn Phi Nn Nn Nn Phi
14	12, 13	syl5eqr 2399	. . . . 5 Nn Phi Nn Phi
15		phiun 4614	. . . . . 6 Phi Nn Phi Nn Phi
16		incom 3448	. . . . . . . . 9 Nn Nn Nn Nn
17		disjdif 3622	. . . . . . . . 9 Nn Nn
18	16, 17	eqtri 2373	. . . . . . . 8 Nn Nn
19		phidisjnn 4615	. . . . . . . 8 Nn Nn Phi Nn Nn
20	18, 19	ax-mp 8	. . . . . . 7 Phi Nn Nn
21	20	uneq1i 3414	. . . . . 6 Phi Nn Phi Nn Phi
22	15, 21	eqtri 2373	. . . . 5 Phi Nn Nn Phi
23	14, 22	syl6eqr 2403	. . . 4 Nn Phi Phi Nn
24	5, 6	difex 4107	. . . . . 6 Nn
25		vex 2862	. . . . . 6
26	24, 25	unex 4106	. . . . 5 Nn
27		phieq 4570	. . . . . 6 Nn Phi Phi Nn
28	27	eqeq2d 2364	. . . . 5 Nn Phi Phi Nn
29	26, 28	spcev 2946	. . . 4 Phi Nn Phi
30	23, 29	syl 15	. . 3 Nn Phi Phi
31	30	exlimiv 1634	. 2 Nn Phi Phi
32	9, 31	syl 15	1 0c Phi

Colors of variables: wff setvar class

Syntax hints:   wn 3   wi 4  wex 1541   wceq 1642   wcel 1710  cvv 2859   cdif 3206   cun 3207   cin 3208   wss 3257  c0 3550   Nn cnnc 4373  0cc0c 4374   Phi cphi 4562

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565

This theorem is referenced by:  phiall  4618

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