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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 4445 | . 2 ⊢ ω ∈ V | |
2 | df-ni 7013 | . . 3 ⊢ N = (ω ∖ {∅}) | |
3 | difss 3149 | . . 3 ⊢ (ω ∖ {∅}) ⊆ ω | |
4 | 2, 3 | eqsstri 3079 | . 2 ⊢ N ⊆ ω |
5 | 1, 4 | ssexi 4006 | 1 ⊢ N ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 Vcvv 2641 ∖ cdif 3018 ∅c0 3310 {csn 3474 ωcom 4442 Ncnpi 6981 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-v 2643 df-dif 3023 df-in 3027 df-ss 3034 df-int 3719 df-iom 4443 df-ni 7013 |
This theorem is referenced by: enqex 7069 nqex 7072 enq0ex 7148 nq0ex 7149 |
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