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Theorem 0cnelphi 4597
 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.
StepHypRef Expression
1 0cnsuc 4401 . . . . . 6 (y +c 1c) ≠ 0c
2 df-ne 2518 . . . . . 6 ((y +c 1c) ≠ 0c ↔ ¬ (y +c 1c) = 0c)
31, 2mpbi 199 . . . . 5 ¬ (y +c 1c) = 0c
4 iffalse 3669 . . . . . . . . . . 11 y Nn → if(y Nn , (y +c 1c), y) = y)
54eqeq2d 2364 . . . . . . . . . 10 y Nn → (0c = if(y Nn , (y +c 1c), y) ↔ 0c = y))
65biimpac 472 . . . . . . . . 9 ((0c = if(y Nn , (y +c 1c), y) ¬ y Nn ) → 0c = y)
7 peano1 4402 . . . . . . . . 9 0c Nn
86, 7syl6eqelr 2442 . . . . . . . 8 ((0c = if(y Nn , (y +c 1c), y) ¬ y Nn ) → y Nn )
98ex 423 . . . . . . 7 (0c = if(y Nn , (y +c 1c), y) → (¬ y Nny Nn ))
109pm2.18d 103 . . . . . 6 (0c = if(y Nn , (y +c 1c), y) → y Nn )
11 iftrue 3668 . . . . . . . . 9 (y Nn → if(y Nn , (y +c 1c), y) = (y +c 1c))
1211eqeq2d 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)
1412, 13syl6bb 252 . . . . . . 7 (y Nn → (0c = if(y Nn , (y +c 1c), y) ↔ (y +c 1c) = 0c))
1514biimpd 198 . . . . . 6 (y Nn → (0c = if(y Nn , (y +c 1c), y) → (y +c 1c) = 0c))
1610, 15mpcom 32 . . . . 5 (0c = if(y Nn , (y +c 1c), y) → (y +c 1c) = 0c)
173, 16mto 167 . . . 4 ¬ 0c = if(y Nn , (y +c 1c), y)
1817a1i 10 . . 3 (y A → ¬ 0c = if(y Nn , (y +c 1c), y))
1918nrex 2716 . 2 ¬ y A 0c = if(y Nn , (y +c 1c), y)
20 0cex 4392 . . 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)))
2221rexbidv 2635 . . 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 4565 . . 3 Phi A = {x y A x = if(y Nn , (y +c 1c), y)}
2420, 22, 23elab2 2988 . 2 (0c Phi Ay A 0c = if(y Nn , (y +c 1c), y))
2519, 24mtbir 290 1 ¬ 0c Phi A
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ifcif 3662  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   Phi cphi 4562 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 4078  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663  df-sn 3741  df-int 3927  df-1c 4136  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565 This theorem is referenced by:  phi011lem1  4598  proj1op  4600  proj2op  4601  phiall  4618
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