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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

Proof of Theorem 0cnelphi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cnsuc 4401 . . . . . 6 1c 0c
2 df-ne 2518 . . . . . 6 1c 0c 1c 0c
31, 2mpbi 199 . . . . 5 1c 0c
4 iffalse 3669 . . . . . . . . . . 11 Nn Nn 1c
54eqeq2d 2364 . . . . . . . . . 10 Nn 0c Nn 1c 0c
65biimpac 472 . . . . . . . . 9 0c Nn 1c Nn 0c
7 peano1 4402 . . . . . . . . 9 0c Nn
86, 7syl6eqelr 2442 . . . . . . . 8 0c Nn 1c Nn Nn
98ex 423 . . . . . . 7 0c Nn 1c Nn Nn
109pm2.18d 103 . . . . . 6 0c Nn 1c Nn
11 iftrue 3668 . . . . . . . . 9 Nn Nn 1c 1c
1211eqeq2d 2364 . . . . . . . 8 Nn 0c Nn 1c 0c 1c
13 eqcom 2355 . . . . . . . 8 0c 1c 1c 0c
1412, 13syl6bb 252 . . . . . . 7 Nn 0c Nn 1c 1c 0c
1514biimpd 198 . . . . . 6 Nn 0c Nn 1c 1c 0c
1610, 15mpcom 32 . . . . 5 0c Nn 1c 1c 0c
173, 16mto 167 . . . 4 0c Nn 1c
1817a1i 10 . . 3 0c Nn 1c
1918nrex 2716 . 2 0c Nn 1c
20 0cex 4392 . . 3 0c
21 eqeq1 2359 . . . 4 0c Nn 1c 0c Nn 1c
2221rexbidv 2635 . . 3 0c Nn 1c 0c Nn 1c
23 df-phi 4565 . . 3 Phi Nn 1c
2420, 22, 23elab2 2988 . 2 0c Phi 0c Nn 1c
2519, 24mtbir 290 1 0c Phi
 Colors of variables: wff setvar class Syntax hints:   wn 3   wa 358   wceq 1642   wcel 1710   wne 2516  wrex 2615  cif 3662  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   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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