Theorem phialllem2 4618

Description: Lemma for phiall 4619. Any set without 0_c is equal to the Phi of a set. (Contributed by Scott Fenton, 8-Apr-2021.)

Hypothesis

Ref	Expression
phiall.1	⊢ A ∈ V

Assertion

Ref	Expression
phialllem2	⊢ (¬ 0c ∈ A → ∃x A = Phi x)

Distinct variable group:   x,A

Proof of Theorem phialllem2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3477	. . 3 ⊢ (A ∩ Nn ) ⊆ Nn
2		inss1 3476	. . . . 5 ⊢ (A ∩ Nn ) ⊆ A
3	2	sseli 3270	. . . 4 ⊢ (0c ∈ (A ∩ Nn ) → 0c ∈ A)
4	3	con3i 127	. . 3 ⊢ (¬ 0c ∈ A → ¬ 0c ∈ (A ∩ Nn ))
5		phiall.1	. . . . 5 ⊢ A ∈ V
6		nncex 4397	. . . . 5 ⊢ Nn ∈ V
7	5, 6	inex 4106	. . . 4 ⊢ (A ∩ Nn ) ∈ V
8	7	phialllem1 4617	. . 3 ⊢ (((A ∩ Nn ) ⊆ Nn ∧ ¬ 0c ∈ (A ∩ Nn )) → ∃y(A ∩ Nn ) = Phi y)
9	1, 4, 8	sylancr 644	. 2 ⊢ (¬ 0c ∈ A → ∃y(A ∩ Nn ) = Phi y)
10		uncom 3409	. . . . . . 7 ⊢ ((A ∖ Nn ) ∪ (A ∩ Nn )) = ((A ∩ Nn ) ∪ (A ∖ Nn ))
11		inundif 3629	. . . . . . 7 ⊢ ((A ∩ Nn ) ∪ (A ∖ Nn )) = A
12	10, 11	eqtri 2373	. . . . . 6 ⊢ ((A ∖ Nn ) ∪ (A ∩ Nn )) = A
13		uneq2 3413	. . . . . 6 ⊢ ((A ∩ Nn ) = Phi y → ((A ∖ Nn ) ∪ (A ∩ Nn )) = ((A ∖ Nn ) ∪ Phi y))
14	12, 13	syl5eqr 2399	. . . . 5 ⊢ ((A ∩ Nn ) = Phi y → A = ((A ∖ Nn ) ∪ Phi y))
15		phiun 4615	. . . . . 6 ⊢ Phi ((A ∖ Nn ) ∪ y) = ( Phi (A ∖ Nn ) ∪ Phi y)
16		incom 3449	. . . . . . . . 9 ⊢ ((A ∖ Nn ) ∩ Nn ) = ( Nn ∩ (A ∖ Nn ))
17		disjdif 3623	. . . . . . . . 9 ⊢ ( Nn ∩ (A ∖ Nn )) = ∅
18	16, 17	eqtri 2373	. . . . . . . 8 ⊢ ((A ∖ Nn ) ∩ Nn ) = ∅
19		phidisjnn 4616	. . . . . . . 8 ⊢ (((A ∖ Nn ) ∩ Nn ) = ∅ → Phi (A ∖ Nn ) = (A ∖ Nn ))
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ Phi (A ∖ Nn ) = (A ∖ Nn )
21	20	uneq1i 3415	. . . . . 6 ⊢ ( Phi (A ∖ Nn ) ∪ Phi y) = ((A ∖ Nn ) ∪ Phi y)
22	15, 21	eqtri 2373	. . . . 5 ⊢ Phi ((A ∖ Nn ) ∪ y) = ((A ∖ Nn ) ∪ Phi y)
23	14, 22	syl6eqr 2403	. . . 4 ⊢ ((A ∩ Nn ) = Phi y → A = Phi ((A ∖ Nn ) ∪ y))
24	5, 6	difex 4108	. . . . . 6 ⊢ (A ∖ Nn ) ∈ V
25		vex 2863	. . . . . 6 ⊢ y ∈ V
26	24, 25	unex 4107	. . . . 5 ⊢ ((A ∖ Nn ) ∪ y) ∈ V
27		phieq 4571	. . . . . 6 ⊢ (x = ((A ∖ Nn ) ∪ y) → Phi x = Phi ((A ∖ Nn ) ∪ y))
28	27	eqeq2d 2364	. . . . 5 ⊢ (x = ((A ∖ Nn ) ∪ y) → (A = Phi x ↔ A = Phi ((A ∖ Nn ) ∪ y)))
29	26, 28	spcev 2947	. . . 4 ⊢ (A = Phi ((A ∖ Nn ) ∪ y) → ∃x A = Phi x)
30	23, 29	syl 15	. . 3 ⊢ ((A ∩ Nn ) = Phi y → ∃x A = Phi x)
31	30	exlimiv 1634	. 2 ⊢ (∃y(A ∩ Nn ) = Phi y → ∃x A = Phi x)
32	9, 31	syl 15	1 ⊢ (¬ 0c ∈ A → ∃x A = Phi x)

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551   Nn cnnc 4374  0_cc0c 4375   Phi cphi 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380  df-phi 4566

This theorem is referenced by:  phiall  4619