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Mirrors > Home > NFE Home > Th. List > phialllem2 | Unicode version |
Description: Lemma for phiall 4618. Any set without 0c is equal to the Phi of a set. (Contributed by Scott Fenton, 8-Apr-2021.) |
Ref | Expression |
---|---|
phiall.1 |
Ref | Expression |
---|---|
phialllem2 | 0c Phi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3476 | . . 3 Nn Nn | |
2 | inss1 3475 | . . . . 5 Nn | |
3 | 2 | sseli 3269 | . . . 4 0c Nn 0c |
4 | 3 | con3i 127 | . . 3 0c 0c Nn |
5 | phiall.1 | . . . . 5 | |
6 | nncex 4396 | . . . . 5 Nn | |
7 | 5, 6 | inex 4105 | . . . 4 Nn |
8 | 7 | phialllem1 4616 | . . 3 Nn Nn 0c Nn Nn Phi |
9 | 1, 4, 8 | sylancr 644 | . 2 0c Nn Phi |
10 | uncom 3408 | . . . . . . 7 Nn Nn Nn Nn | |
11 | inundif 3628 | . . . . . . 7 Nn Nn | |
12 | 10, 11 | eqtri 2373 | . . . . . 6 Nn Nn |
13 | uneq2 3412 | . . . . . 6 Nn Phi Nn Nn Nn Phi | |
14 | 12, 13 | syl5eqr 2399 | . . . . 5 Nn Phi Nn Phi |
15 | phiun 4614 | . . . . . 6 Phi Nn Phi Nn Phi | |
16 | incom 3448 | . . . . . . . . 9 Nn Nn Nn Nn | |
17 | disjdif 3622 | . . . . . . . . 9 Nn Nn | |
18 | 16, 17 | eqtri 2373 | . . . . . . . 8 Nn Nn |
19 | phidisjnn 4615 | . . . . . . . 8 Nn Nn Phi Nn Nn | |
20 | 18, 19 | ax-mp 5 | . . . . . . 7 Phi Nn Nn |
21 | 20 | uneq1i 3414 | . . . . . 6 Phi Nn Phi Nn Phi |
22 | 15, 21 | eqtri 2373 | . . . . 5 Phi Nn Nn Phi |
23 | 14, 22 | syl6eqr 2403 | . . . 4 Nn Phi Phi Nn |
24 | 5, 6 | difex 4107 | . . . . . 6 Nn |
25 | vex 2862 | . . . . . 6 | |
26 | 24, 25 | unex 4106 | . . . . 5 Nn |
27 | phieq 4570 | . . . . . 6 Nn Phi Phi Nn | |
28 | 27 | eqeq2d 2364 | . . . . 5 Nn Phi Phi Nn |
29 | 26, 28 | spcev 2946 | . . . 4 Phi Nn Phi |
30 | 23, 29 | syl 15 | . . 3 Nn Phi Phi |
31 | 30 | exlimiv 1634 | . 2 Nn Phi Phi |
32 | 9, 31 | syl 15 | 1 0c Phi |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wex 1541 wceq 1642 wcel 1710 cvv 2859 cdif 3206 cun 3207 cin 3208 wss 3257 c0 3550 Nn cnnc 4373 0cc0c 4374 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-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: phiall 4618 |
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