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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 3592 | . . . . . . . . . 10 Nn Nn | |
2 | 1 | biimpi 186 | . . . . . . . . 9 Nn Nn |
3 | 2 | r19.21bi 2713 | . . . . . . . 8 Nn Nn |
4 | iffalse 3670 | . . . . . . . 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 2632 | . . . 4 Nn Nn 1c |
10 | risset 2662 | . . . 4 | |
11 | 9, 10 | syl6bbr 254 | . . 3 Nn Nn 1c |
12 | 11 | alrimiv 1631 | . 2 Nn Nn 1c |
13 | df-phi 4566 | . . . 4 Phi Nn 1c | |
14 | 13 | eqeq1i 2360 | . . 3 Phi Nn 1c |
15 | abeq1 2460 | . . 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 2615 wrex 2616 cin 3209 c0 3551 cif 3663 1cc1c 4135 Nn cnnc 4374 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 |
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-dif 3216 df-nul 3552 df-if 3664 df-phi 4566 |
This theorem is referenced by: phialllem2 4618 |
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