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Mirrors > Home > ILE Home > Th. List > niex | GIF version |
Description: The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
Ref | Expression |
---|---|
niex | ⊢ N ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4507 | . 2 ⊢ ω ∈ V | |
2 | df-ni 7112 | . . 3 ⊢ N = (ω ∖ {∅}) | |
3 | difss 3202 | . . 3 ⊢ (ω ∖ {∅}) ⊆ ω | |
4 | 2, 3 | eqsstri 3129 | . 2 ⊢ N ⊆ ω |
5 | 1, 4 | ssexi 4066 | 1 ⊢ N ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 ∖ cdif 3068 ∅c0 3363 {csn 3527 ωcom 4504 Ncnpi 7080 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-iinf 4502 |
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-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-int 3772 df-iom 4505 df-ni 7112 |
This theorem is referenced by: enqex 7168 nqex 7171 enq0ex 7247 nq0ex 7248 |
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