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Mirrors > Home > ILE Home > Th. List > pp0ex | GIF version |
Description: {∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
pp0ex | ⊢ {∅, {∅}} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p0ex 4021 | . . 3 ⊢ {∅} ∈ V | |
2 | 1 | pwex 4016 | . 2 ⊢ 𝒫 {∅} ∈ V |
3 | pwpw0ss 3646 | . 2 ⊢ {∅, {∅}} ⊆ 𝒫 {∅} | |
4 | 2, 3 | ssexi 3975 | 1 ⊢ {∅, {∅}} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 Vcvv 2619 ∅c0 3286 𝒫 cpw 3427 {csn 3444 {cpr 3445 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-nul 3963 ax-pow 4007 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3429 df-sn 3450 df-pr 3451 |
This theorem is referenced by: ord3ex 4023 ontr2exmid 4339 ordtri2or2exmidlem 4340 onsucelsucexmidlem 4343 regexmid 4349 reg2exmid 4350 reg3exmid 4393 nnregexmid 4432 acexmidlemcase 5639 acexmidlemv 5642 |
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