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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 8267 | . 2 ⊢ 1o ∈ ω | |
2 | 1n0 8121 | . 2 ⊢ 1o ≠ ∅ | |
3 | elni 10300 | . 2 ⊢ (1o ∈ N ↔ (1o ∈ ω ∧ 1o ≠ ∅)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ 1o ∈ N |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 ωcom 7582 1oc1o 8097 Ncnpi 10268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-om 7583 df-1o 8104 df-ni 10296 |
This theorem is referenced by: mulidpi 10310 1lt2pi 10329 nlt1pi 10330 indpi 10331 pinq 10351 1nq 10352 1nqenq 10386 mulidnq 10387 1lt2nq 10397 archnq 10404 prlem934 10457 |
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