Theorem phiall 4619

Description: Any set is equal to either the Phi of another set or to a Phi with 0_c adjoined. (Contributed by Scott Fenton, 8-Apr-2021.)

Hypothesis

Ref	Expression
phiall.1	⊢ A ∈ V

Assertion

Ref	Expression
phiall	⊢ ∃x(A = Phi x ∨ A = ( Phi x ∪ {0c}))

Distinct variable group:   x,A

Proof of Theorem phiall

Step	Hyp	Ref	Expression
1		neldifsn 3842	. . . . 5 ⊢ ¬ 0c ∈ (A ∖ {0c})
2		phiall.1	. . . . . . 7 ⊢ A ∈ V
3		snex 4112	. . . . . . 7 ⊢ {0c} ∈ V
4	2, 3	difex 4108	. . . . . 6 ⊢ (A ∖ {0c}) ∈ V
5	4	phialllem2 4618	. . . . 5 ⊢ (¬ 0c ∈ (A ∖ {0c}) → ∃x(A ∖ {0c}) = Phi x)
6	1, 5	ax-mp 5	. . . 4 ⊢ ∃x(A ∖ {0c}) = Phi x
7		disjsn 3787	. . . . . . . . . 10 ⊢ (((A ∖ {0c}) ∩ {0c}) = ∅ ↔ ¬ 0c ∈ (A ∖ {0c}))
8	1, 7	mpbir 200	. . . . . . . . 9 ⊢ ((A ∖ {0c}) ∩ {0c}) = ∅
9		0cnelphi 4598	. . . . . . . . . 10 ⊢ ¬ 0c ∈ Phi x
10		disjsn 3787	. . . . . . . . . 10 ⊢ (( Phi x ∩ {0c}) = ∅ ↔ ¬ 0c ∈ Phi x)
11	9, 10	mpbir 200	. . . . . . . . 9 ⊢ ( Phi x ∩ {0c}) = ∅
12	8, 11	eqtr4i 2376	. . . . . . . 8 ⊢ ((A ∖ {0c}) ∩ {0c}) = ( Phi x ∩ {0c})
13	12	biantru 491	. . . . . . 7 ⊢ (((A ∖ {0c}) ∪ {0c}) = ( Phi x ∪ {0c}) ↔ (((A ∖ {0c}) ∪ {0c}) = ( Phi x ∪ {0c}) ∧ ((A ∖ {0c}) ∩ {0c}) = ( Phi x ∩ {0c})))
14		unineq 3506	. . . . . . 7 ⊢ ((((A ∖ {0c}) ∪ {0c}) = ( Phi x ∪ {0c}) ∧ ((A ∖ {0c}) ∩ {0c}) = ( Phi x ∩ {0c})) ↔ (A ∖ {0c}) = Phi x)
15	13, 14	bitri 240	. . . . . 6 ⊢ (((A ∖ {0c}) ∪ {0c}) = ( Phi x ∪ {0c}) ↔ (A ∖ {0c}) = Phi x)
16		difsnid 3855	. . . . . . 7 ⊢ (0c ∈ A → ((A ∖ {0c}) ∪ {0c}) = A)
17	16	eqeq1d 2361	. . . . . 6 ⊢ (0c ∈ A → (((A ∖ {0c}) ∪ {0c}) = ( Phi x ∪ {0c}) ↔ A = ( Phi x ∪ {0c})))
18	15, 17	syl5bbr 250	. . . . 5 ⊢ (0c ∈ A → ((A ∖ {0c}) = Phi x ↔ A = ( Phi x ∪ {0c})))
19	18	exbidv 1626	. . . 4 ⊢ (0c ∈ A → (∃x(A ∖ {0c}) = Phi x ↔ ∃x A = ( Phi x ∪ {0c})))
20	6, 19	mpbii 202	. . 3 ⊢ (0c ∈ A → ∃x A = ( Phi x ∪ {0c}))
21		olc 373	. . . 4 ⊢ (A = ( Phi x ∪ {0c}) → (A = Phi x ∨ A = ( Phi x ∪ {0c})))
22	21	eximi 1576	. . 3 ⊢ (∃x A = ( Phi x ∪ {0c}) → ∃x(A = Phi x ∨ A = ( Phi x ∪ {0c})))
23	20, 22	syl 15	. 2 ⊢ (0c ∈ A → ∃x(A = Phi x ∨ A = ( Phi x ∪ {0c})))
24	2	phialllem2 4618	. . 3 ⊢ (¬ 0c ∈ A → ∃x A = Phi x)
25		orc 374	. . . 4 ⊢ (A = Phi x → (A = Phi x ∨ A = ( Phi x ∪ {0c})))
26	25	eximi 1576	. . 3 ⊢ (∃x A = Phi x → ∃x(A = Phi x ∨ A = ( Phi x ∪ {0c})))
27	24, 26	syl 15	. 2 ⊢ (¬ 0c ∈ A → ∃x(A = Phi x ∨ A = ( Phi x ∪ {0c})))
28	23, 27	pm2.61i 156	1 ⊢ ∃x(A = Phi x ∨ A = ( Phi x ∪ {0c}))

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  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:  opeq  4620