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Mirrors > Home > NFE Home > Th. List > phiall | Unicode version |
Description: Any set is equal to either the Phi of another set or to a Phi with 0c adjoined. (Contributed by Scott Fenton, 8-Apr-2021.) |
Ref | Expression |
---|---|
phiall.1 |
Ref | Expression |
---|---|
phiall | Phi Phi 0c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neldifsn 3841 | . . . . 5 0c 0c | |
2 | phiall.1 | . . . . . . 7 | |
3 | snex 4111 | . . . . . . 7 0c | |
4 | 2, 3 | difex 4107 | . . . . . 6 0c |
5 | 4 | phialllem2 4617 | . . . . 5 0c 0c 0c Phi |
6 | 1, 5 | ax-mp 8 | . . . 4 0c Phi |
7 | disjsn 3786 | . . . . . . . . . 10 0c 0c 0c 0c | |
8 | 1, 7 | mpbir 200 | . . . . . . . . 9 0c 0c |
9 | 0cnelphi 4597 | . . . . . . . . . 10 0c Phi | |
10 | disjsn 3786 | . . . . . . . . . 10 Phi 0c 0c Phi | |
11 | 9, 10 | mpbir 200 | . . . . . . . . 9 Phi 0c |
12 | 8, 11 | eqtr4i 2376 | . . . . . . . 8 0c 0c Phi 0c |
13 | 12 | biantru 491 | . . . . . . 7 0c 0c Phi 0c 0c 0c Phi 0c 0c 0c Phi 0c |
14 | unineq 3505 | . . . . . . 7 0c 0c Phi 0c 0c 0c Phi 0c 0c Phi | |
15 | 13, 14 | bitri 240 | . . . . . 6 0c 0c Phi 0c 0c Phi |
16 | difsnid 3854 | . . . . . . 7 0c 0c 0c | |
17 | 16 | eqeq1d 2361 | . . . . . 6 0c 0c 0c Phi 0c Phi 0c |
18 | 15, 17 | syl5bbr 250 | . . . . 5 0c 0c Phi Phi 0c |
19 | 18 | exbidv 1626 | . . . 4 0c 0c Phi Phi 0c |
20 | 6, 19 | mpbii 202 | . . 3 0c Phi 0c |
21 | olc 373 | . . . 4 Phi 0c Phi Phi 0c | |
22 | 21 | eximi 1576 | . . 3 Phi 0c Phi Phi 0c |
23 | 20, 22 | syl 15 | . 2 0c Phi Phi 0c |
24 | 2 | phialllem2 4617 | . . 3 0c Phi |
25 | orc 374 | . . . 4 Phi Phi Phi 0c | |
26 | 25 | eximi 1576 | . . 3 Phi Phi Phi 0c |
27 | 24, 26 | syl 15 | . 2 0c Phi Phi 0c |
28 | 23, 27 | pm2.61i 156 | 1 Phi Phi 0c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wo 357 wa 358 wex 1541 wceq 1642 wcel 1710 cvv 2859 cdif 3206 cun 3207 cin 3208 c0 3550 csn 3737 0cc0c 4374 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 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 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-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-0c 4377 df-addc 4378 df-nnc 4379 df-phi 4565 |
This theorem is referenced by: opeq 4619 |
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