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Mirrors > Home > MPE Home > Th. List > 1pi | Structured version Visualization version GIF version |
Description: Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1pi | ⊢ 1o ∈ N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8661 | . 2 ⊢ 1o ∈ ω | |
2 | 1n0 8509 | . 2 ⊢ 1o ≠ ∅ | |
3 | elni 10901 | . 2 ⊢ (1o ∈ N ↔ (1o ∈ ω ∧ 1o ≠ ∅)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ 1o ∈ N |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ≠ wne 2929 ∅c0 4322 ωcom 7871 1oc1o 8480 Ncnpi 10869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-om 7872 df-1o 8487 df-ni 10897 |
This theorem is referenced by: mulidpi 10911 1lt2pi 10930 nlt1pi 10931 indpi 10932 pinq 10952 1nq 10953 1nqenq 10987 mulidnq 10988 1lt2nq 10998 archnq 11005 prlem934 11058 |
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