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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw0 3738 |
. 2
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2 | 0ex 4127 |
. . 3
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3 | 2 | pwex 4180 |
. 2
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4 | 1, 3 | eqeltrri 2251 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 |
This theorem is referenced by: pp0ex 4186 undifexmid 4190 exmidexmid 4193 exmidundif 4203 exmidundifim 4204 exmid1stab 4205 ordtriexmidlem 4515 ontr2exmid 4521 onsucsssucexmid 4523 onsucelsucexmid 4526 regexmidlemm 4528 ordsoexmid 4558 ordtri2or2exmid 4567 ontri2orexmidim 4568 opthprc 4674 acexmidlema 5860 acexmidlem2 5866 tposexg 6253 2dom 6799 map1 6806 endisj 6818 ssfiexmid 6870 domfiexmid 6872 exmidpw 6902 djuex 7036 exmidomni 7134 exmidonfinlem 7186 exmidfodomrlemr 7195 exmidfodomrlemrALT 7196 exmidaclem 7201 pw1dom2 7220 pw1ne1 7222 sbthom 14430 |
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