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Mirrors > Home > NFE Home > Th. List > phidisjnn | Unicode version |
Description: The phi operation applied to a set disjoint from the naturals has no effect. (Contributed by SF, 20-Feb-2015.) |
Ref | Expression |
---|---|
phidisjnn | Nn Phi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj 3591 | . . . . . . . . . 10 Nn Nn | |
2 | 1 | biimpi 186 | . . . . . . . . 9 Nn Nn |
3 | 2 | r19.21bi 2712 | . . . . . . . 8 Nn Nn |
4 | iffalse 3669 | . . . . . . . 8 Nn Nn 1c | |
5 | 3, 4 | syl 15 | . . . . . . 7 Nn Nn 1c |
6 | 5 | eqeq2d 2364 | . . . . . 6 Nn Nn 1c |
7 | equcom 1680 | . . . . . 6 | |
8 | 6, 7 | syl6bbr 254 | . . . . 5 Nn Nn 1c |
9 | 8 | rexbidva 2631 | . . . 4 Nn Nn 1c |
10 | risset 2661 | . . . 4 | |
11 | 9, 10 | syl6bbr 254 | . . 3 Nn Nn 1c |
12 | 11 | alrimiv 1631 | . 2 Nn Nn 1c |
13 | df-phi 4565 | . . . 4 Phi Nn 1c | |
14 | 13 | eqeq1i 2360 | . . 3 Phi Nn 1c |
15 | abeq1 2459 | . . 3 Nn 1c Nn 1c | |
16 | 14, 15 | bitri 240 | . 2 Phi Nn 1c |
17 | 12, 16 | sylibr 203 | 1 Nn Phi |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wal 1540 wceq 1642 wcel 1710 cab 2339 wral 2614 wrex 2615 cin 3208 c0 3550 cif 3662 1cc1c 4134 Nn cnnc 4373 cplc 4375 Phi cphi 4562 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 |
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-dif 3215 df-nul 3551 df-if 3663 df-phi 4565 |
This theorem is referenced by: phialllem2 4617 |
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