Theorem 0cnelphi 4598

Description: Cardinal zero is not a member of a phi operation. Theorem X.2.3 of [Rosser] p. 282. (Contributed by SF, 3-Feb-2015.)

Assertion

Ref	Expression
0cnelphi	⊢ ¬ 0c ∈ Phi A

Proof of Theorem 0cnelphi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cnsuc 4402	. . . . . 6 ⊢ (y +c 1c) ≠ 0c
2		df-ne 2519	. . . . . 6 ⊢ ((y +c 1c) ≠ 0c ↔ ¬ (y +c 1c) = 0c)
3	1, 2	mpbi 199	. . . . 5 ⊢ ¬ (y +c 1c) = 0c
4		iffalse 3670	. . . . . . . . . . 11 ⊢ (¬ y ∈ Nn → if(y ∈ Nn , (y +c 1c), y) = y)
5	4	eqeq2d 2364	. . . . . . . . . 10 ⊢ (¬ y ∈ Nn → (0c = if(y ∈ Nn , (y +c 1c), y) ↔ 0c = y))
6	5	biimpac 472	. . . . . . . . 9 ⊢ ((0c = if(y ∈ Nn , (y +c 1c), y) ∧ ¬ y ∈ Nn ) → 0c = y)
7		peano1 4403	. . . . . . . . 9 ⊢ 0c ∈ Nn
8	6, 7	syl6eqelr 2442	. . . . . . . 8 ⊢ ((0c = if(y ∈ Nn , (y +c 1c), y) ∧ ¬ y ∈ Nn ) → y ∈ Nn )
9	8	ex 423	. . . . . . 7 ⊢ (0c = if(y ∈ Nn , (y +c 1c), y) → (¬ y ∈ Nn → y ∈ Nn ))
10	9	pm2.18d 103	. . . . . 6 ⊢ (0c = if(y ∈ Nn , (y +c 1c), y) → y ∈ Nn )
11		iftrue 3669	. . . . . . . . 9 ⊢ (y ∈ Nn → if(y ∈ Nn , (y +c 1c), y) = (y +c 1c))
12	11	eqeq2d 2364	. . . . . . . 8 ⊢ (y ∈ Nn → (0c = if(y ∈ Nn , (y +c 1c), y) ↔ 0c = (y +c 1c)))
13		eqcom 2355	. . . . . . . 8 ⊢ (0c = (y +c 1c) ↔ (y +c 1c) = 0c)
14	12, 13	syl6bb 252	. . . . . . 7 ⊢ (y ∈ Nn → (0c = if(y ∈ Nn , (y +c 1c), y) ↔ (y +c 1c) = 0c))
15	14	biimpd 198	. . . . . 6 ⊢ (y ∈ Nn → (0c = if(y ∈ Nn , (y +c 1c), y) → (y +c 1c) = 0c))
16	10, 15	mpcom 32	. . . . 5 ⊢ (0c = if(y ∈ Nn , (y +c 1c), y) → (y +c 1c) = 0c)
17	3, 16	mto 167	. . . 4 ⊢ ¬ 0c = if(y ∈ Nn , (y +c 1c), y)
18	17	a1i 10	. . 3 ⊢ (y ∈ A → ¬ 0c = if(y ∈ Nn , (y +c 1c), y))
19	18	nrex 2717	. 2 ⊢ ¬ ∃y ∈ A 0c = if(y ∈ Nn , (y +c 1c), y)
20		0cex 4393	. . 3 ⊢ 0c ∈ V
21		eqeq1 2359	. . . 4 ⊢ (x = 0c → (x = if(y ∈ Nn , (y +c 1c), y) ↔ 0c = if(y ∈ Nn , (y +c 1c), y)))
22	21	rexbidv 2636	. . 3 ⊢ (x = 0c → (∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y) ↔ ∃y ∈ A 0c = if(y ∈ Nn , (y +c 1c), y)))
23		df-phi 4566	. . 3 ⊢ Phi A = {x ∣ ∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y)}
24	20, 22, 23	elab2 2989	. 2 ⊢ (0c ∈ Phi A ↔ ∃y ∈ A 0c = if(y ∈ Nn , (y +c 1c), y))
25	19, 24	mtbir 290	1 ⊢ ¬ 0c ∈ Phi A

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616   ifcif 3663  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   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-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-if 3664  df-sn 3742  df-int 3928  df-1c 4137  df-0c 4378  df-addc 4379  df-nnc 4380  df-phi 4566

This theorem is referenced by:  phi011lem1  4599  proj1op  4601  proj2op  4602  phiall  4619