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Mirrors > Home > ILE Home > Th. List > p0ex | Unicode version |
Description: The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.) |
Ref | Expression |
---|---|
p0ex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw0 3714 | . 2 | |
2 | 0ex 4103 | . . 3 | |
3 | 2 | pwex 4156 | . 2 |
4 | 1, 3 | eqeltrri 2238 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 2135 cvv 2721 c0 3404 cpw 3553 csn 3570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 |
This theorem is referenced by: pp0ex 4162 undifexmid 4166 exmidexmid 4169 exmidundif 4179 exmidundifim 4180 ordtriexmidlem 4490 ontr2exmid 4496 onsucsssucexmid 4498 onsucelsucexmid 4501 regexmidlemm 4503 ordsoexmid 4533 ordtri2or2exmid 4542 ontri2orexmidim 4543 opthprc 4649 acexmidlema 5827 acexmidlem2 5833 tposexg 6217 2dom 6762 map1 6769 endisj 6781 ssfiexmid 6833 domfiexmid 6835 exmidpw 6865 djuex 6999 exmidomni 7097 exmidonfinlem 7140 exmidfodomrlemr 7149 exmidfodomrlemrALT 7150 exmidaclem 7155 pw1dom2 7174 pw1ne1 7176 exmid1stab 13721 sbthom 13746 |
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