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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 7877 | . 2 ⊢ 1𝑜 ∈ ω | |
2 | 1n0 7733 | . 2 ⊢ 1𝑜 ≠ ∅ | |
3 | elni 9904 | . 2 ⊢ (1𝑜 ∈ N ↔ (1𝑜 ∈ ω ∧ 1𝑜 ≠ ∅)) | |
4 | 1, 2, 3 | mpbir2an 690 | 1 ⊢ 1𝑜 ∈ N |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 ≠ wne 2943 ∅c0 4063 ωcom 7216 1𝑜c1o 7710 Ncnpi 9872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-tr 4888 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-om 7217 df-1o 7717 df-ni 9900 |
This theorem is referenced by: mulidpi 9914 1lt2pi 9933 nlt1pi 9934 indpi 9935 pinq 9955 1nq 9956 1nqenq 9990 mulidnq 9991 1lt2nq 10001 archnq 10008 prlem934 10061 |
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