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Mirrors > Home > NFE Home > Th. List > pw1equn | Unicode version |
Description: A condition for a unit power class to equal a union. (Contributed by SF, 26-Jan-2015.) |
Ref | Expression |
---|---|
pw1equn.1 | |
pw1equn.2 |
Ref | Expression |
---|---|
pw1equn | 1 1 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unipw1 4325 | . . . 4 1 | |
2 | unieq 3900 | . . . 4 1 1 | |
3 | 1, 2 | syl5eqr 2399 | . . 3 1 |
4 | ssun1 3426 | . . . . . 6 | |
5 | sseq2 3293 | . . . . . 6 1 1 | |
6 | 4, 5 | mpbiri 224 | . . . . 5 1 1 |
7 | pw1ss1c 4158 | . . . . 5 1 1c | |
8 | 6, 7 | syl6ss 3284 | . . . 4 1 1c |
9 | eqpw1uni 4330 | . . . 4 1c 1 | |
10 | 8, 9 | syl 15 | . . 3 1 1 |
11 | ssun2 3427 | . . . . . 6 | |
12 | sseq2 3293 | . . . . . 6 1 1 | |
13 | 11, 12 | mpbiri 224 | . . . . 5 1 1 |
14 | 13, 7 | syl6ss 3284 | . . . 4 1 1c |
15 | eqpw1uni 4330 | . . . 4 1c 1 | |
16 | 14, 15 | syl 15 | . . 3 1 1 |
17 | pw1equn.1 | . . . . 5 | |
18 | 17 | uniex 4317 | . . . 4 |
19 | pw1equn.2 | . . . . 5 | |
20 | 19 | uniex 4317 | . . . 4 |
21 | uneq12 3413 | . . . . . . 7 | |
22 | uniun 3910 | . . . . . . 7 | |
23 | 21, 22 | syl6eqr 2403 | . . . . . 6 |
24 | 23 | eqeq2d 2364 | . . . . 5 |
25 | pw1eq 4143 | . . . . . . 7 1 1 | |
26 | 25 | eqeq2d 2364 | . . . . . 6 1 1 |
27 | 26 | adantr 451 | . . . . 5 1 1 |
28 | pw1eq 4143 | . . . . . . 7 1 1 | |
29 | 28 | eqeq2d 2364 | . . . . . 6 1 1 |
30 | 29 | adantl 452 | . . . . 5 1 1 |
31 | 24, 27, 30 | 3anbi123d 1252 | . . . 4 1 1 1 1 |
32 | 18, 20, 31 | spc2ev 2947 | . . 3 1 1 1 1 |
33 | 3, 10, 16, 32 | syl3anc 1182 | . 2 1 1 1 |
34 | pw1un 4163 | . . . 4 1 1 1 | |
35 | pw1eq 4143 | . . . . . 6 1 1 | |
36 | uneq12 3413 | . . . . . 6 1 1 1 1 | |
37 | 35, 36 | eqeqan12d 2368 | . . . . 5 1 1 1 1 1 1 |
38 | 37 | 3impb 1147 | . . . 4 1 1 1 1 1 1 |
39 | 34, 38 | mpbiri 224 | . . 3 1 1 1 |
40 | 39 | exlimivv 1635 | . 2 1 1 1 |
41 | 33, 40 | impbii 180 | 1 1 1 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 w3a 934 wex 1541 wceq 1642 wcel 1710 cvv 2859 cun 3207 wss 3257 cuni 3891 1cc1c 4134 1 cpw1 4135 |
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-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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-imak 4189 df-p6 4191 df-sik 4192 df-ssetk 4193 |
This theorem is referenced by: taddc 6229 letc 6231 |
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