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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 4564 | . 2 ⊢ ω ∈ V | |
2 | df-ni 7236 | . . 3 ⊢ N = (ω ∖ {∅}) | |
3 | difss 3243 | . . 3 ⊢ (ω ∖ {∅}) ⊆ ω | |
4 | 2, 3 | eqsstri 3169 | . 2 ⊢ N ⊆ ω |
5 | 1, 4 | ssexi 4114 | 1 ⊢ N ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 Vcvv 2721 ∖ cdif 3108 ∅c0 3404 {csn 3570 ωcom 4561 Ncnpi 7204 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-int 3819 df-iom 4562 df-ni 7236 |
This theorem is referenced by: enqex 7292 nqex 7295 enq0ex 7371 nq0ex 7372 |
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