Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > p0ex | GIF version |
Description: The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.) |
Ref | Expression |
---|---|
p0ex | ⊢ {∅} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw0 3667 | . 2 ⊢ 𝒫 ∅ = {∅} | |
2 | 0ex 4055 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | pwex 4107 | . 2 ⊢ 𝒫 ∅ ∈ V |
4 | 1, 3 | eqeltrri 2213 | 1 ⊢ {∅} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 ∅c0 3363 𝒫 cpw 3510 {csn 3527 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 |
This theorem is referenced by: pp0ex 4113 undifexmid 4117 exmidexmid 4120 exmidundif 4129 exmidundifim 4130 ordtriexmidlem 4435 ontr2exmid 4440 onsucsssucexmid 4442 onsucelsucexmid 4445 regexmidlemm 4447 ordsoexmid 4477 ordtri2or2exmid 4486 opthprc 4590 acexmidlema 5765 acexmidlem2 5771 tposexg 6155 2dom 6699 map1 6706 endisj 6718 ssfiexmid 6770 domfiexmid 6772 exmidpw 6802 djuex 6928 exmidomni 7014 exmidonfinlem 7049 exmidfodomrlemr 7058 exmidfodomrlemrALT 7059 exmidaclem 7064 pw1dom2 13190 exmid1stab 13195 sbthom 13221 |
Copyright terms: Public domain | W3C validator |