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Mirrors > Home > NFE Home > Th. List > 0cnelphi | Unicode version |
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.) |
Ref | Expression |
---|---|
0cnelphi | 0c Phi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cnsuc 4401 | . . . . . 6 1c 0c | |
2 | df-ne 2518 | . . . . . 6 1c 0c 1c 0c | |
3 | 1, 2 | mpbi 199 | . . . . 5 1c 0c |
4 | iffalse 3669 | . . . . . . . . . . 11 Nn Nn 1c | |
5 | 4 | eqeq2d 2364 | . . . . . . . . . 10 Nn 0c Nn 1c 0c |
6 | 5 | biimpac 472 | . . . . . . . . 9 0c Nn 1c Nn 0c |
7 | peano1 4402 | . . . . . . . . 9 0c Nn | |
8 | 6, 7 | syl6eqelr 2442 | . . . . . . . 8 0c Nn 1c Nn Nn |
9 | 8 | ex 423 | . . . . . . 7 0c Nn 1c Nn Nn |
10 | 9 | pm2.18d 103 | . . . . . 6 0c Nn 1c Nn |
11 | iftrue 3668 | . . . . . . . . 9 Nn Nn 1c 1c | |
12 | 11 | eqeq2d 2364 | . . . . . . . 8 Nn 0c Nn 1c 0c 1c |
13 | eqcom 2355 | . . . . . . . 8 0c 1c 1c 0c | |
14 | 12, 13 | syl6bb 252 | . . . . . . 7 Nn 0c Nn 1c 1c 0c |
15 | 14 | biimpd 198 | . . . . . 6 Nn 0c Nn 1c 1c 0c |
16 | 10, 15 | mpcom 32 | . . . . 5 0c Nn 1c 1c 0c |
17 | 3, 16 | mto 167 | . . . 4 0c Nn 1c |
18 | 17 | a1i 10 | . . 3 0c Nn 1c |
19 | 18 | nrex 2716 | . 2 0c Nn 1c |
20 | 0cex 4392 | . . 3 0c | |
21 | eqeq1 2359 | . . . 4 0c Nn 1c 0c Nn 1c | |
22 | 21 | rexbidv 2635 | . . 3 0c Nn 1c 0c Nn 1c |
23 | df-phi 4565 | . . 3 Phi Nn 1c | |
24 | 20, 22, 23 | elab2 2988 | . 2 0c Phi 0c Nn 1c |
25 | 19, 24 | mtbir 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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