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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 4206 | . . 3 ⊢ {∅} ∈ V | |
2 | 1 | pwex 4201 | . 2 ⊢ 𝒫 {∅} ∈ V |
3 | pwpw0ss 3819 | . 2 ⊢ {∅, {∅}} ⊆ 𝒫 {∅} | |
4 | 2, 3 | ssexi 4156 | 1 ⊢ {∅, {∅}} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 Vcvv 2752 ∅c0 3437 𝒫 cpw 3590 {csn 3607 {cpr 3608 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 |
This theorem is referenced by: ord3ex 4208 ontr2exmid 4542 ordtri2or2exmidlem 4543 onsucelsucexmidlem 4546 regexmid 4552 reg2exmid 4553 reg3exmid 4597 nnregexmid 4638 acexmidlemcase 5892 acexmidlemv 5895 exmidpw2en 6941 exmidaclem 7238 |
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