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Mirrors > Home > NFE Home > Th. List > sspw12 | Unicode version |
Description: A set is a subset of cardinal one iff it is the unit power class of some other set. (Contributed by SF, 17-Mar-2015.) |
Ref | Expression |
---|---|
sspw12.1 |
Ref | Expression |
---|---|
sspw12 | 1c 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqpw1uni 4330 | . . 3 1c 1 | |
2 | sspw12.1 | . . . . 5 | |
3 | 2 | uniex 4317 | . . . 4 |
4 | pw1eq 4143 | . . . . 5 1 1 | |
5 | 4 | eqeq2d 2364 | . . . 4 1 1 |
6 | 3, 5 | spcev 2946 | . . 3 1 1 |
7 | 1, 6 | syl 15 | . 2 1c 1 |
8 | pw1ss1c 4158 | . . . 4 1 1c | |
9 | sseq1 3292 | . . . 4 1 1c 1 1c | |
10 | 8, 9 | mpbiri 224 | . . 3 1 1c |
11 | 10 | exlimiv 1634 | . 2 1 1c |
12 | 7, 11 | impbii 180 | 1 1c 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wex 1541 wceq 1642 wcel 1710 cvv 2859 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: ce0lenc1 6239 |
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