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Mirrors > Home > ILE Home > Th. List > npex | GIF version |
Description: The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
Ref | Expression |
---|---|
npex | ⊢ P ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqex 7164 | . . . 4 ⊢ Q ∈ V | |
2 | 1 | pwex 4102 | . . 3 ⊢ 𝒫 Q ∈ V |
3 | 2, 2 | xpex 4649 | . 2 ⊢ (𝒫 Q × 𝒫 Q) ∈ V |
4 | npsspw 7272 | . 2 ⊢ P ⊆ (𝒫 Q × 𝒫 Q) | |
5 | 3, 4 | ssexi 4061 | 1 ⊢ P ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2681 𝒫 cpw 3505 × cxp 4532 Qcnq 7081 Pcnp 7092 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-qs 6428 df-ni 7105 df-nqqs 7149 df-inp 7267 |
This theorem is referenced by: suplocexprlem2b 7515 suplocexprlemlub 7525 enrex 7538 addvalex 7645 axcnex 7660 |
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