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Mirrors > Home > ILE Home > Th. List > peano1nnnn | Unicode version |
Description: One is an element of . This is a counterpart to 1nn 8878 designed for real number axioms which involve natural numbers (notably, axcaucvg 7851). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
peano1nnnn.n |
Ref | Expression |
---|---|
peano1nnnn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1nnnn.n | . . . 4 | |
2 | 1 | eleq2i 2237 | . . 3 |
3 | df-1 7771 | . . . . 5 | |
4 | 1sr 7702 | . . . . . 6 | |
5 | opelreal 7778 | . . . . . 6 | |
6 | 4, 5 | mpbir 145 | . . . . 5 |
7 | 3, 6 | eqeltri 2243 | . . . 4 |
8 | elintg 3837 | . . . 4 | |
9 | 7, 8 | ax-mp 5 | . . 3 |
10 | 2, 9 | bitri 183 | . 2 |
11 | vex 2733 | . . . 4 | |
12 | eleq2 2234 | . . . . 5 | |
13 | eleq2 2234 | . . . . . 6 | |
14 | 13 | raleqbi1dv 2673 | . . . . 5 |
15 | 12, 14 | anbi12d 470 | . . . 4 |
16 | 11, 15 | elab 2874 | . . 3 |
17 | 16 | simplbi 272 | . 2 |
18 | 10, 17 | mprgbir 2528 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wcel 2141 cab 2156 wral 2448 cop 3584 cint 3829 (class class class)co 5851 cnr 7248 c0r 7249 c1r 7250 cr 7762 c1 7764 caddc 7766 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-1o 6393 df-2o 6394 df-oadd 6397 df-omul 6398 df-er 6510 df-ec 6512 df-qs 6516 df-ni 7255 df-pli 7256 df-mi 7257 df-lti 7258 df-plpq 7295 df-mpq 7296 df-enq 7298 df-nqqs 7299 df-plqqs 7300 df-mqqs 7301 df-1nqqs 7302 df-rq 7303 df-ltnqqs 7304 df-enq0 7375 df-nq0 7376 df-0nq0 7377 df-plq0 7378 df-mq0 7379 df-inp 7417 df-i1p 7418 df-iplp 7419 df-enr 7677 df-nr 7678 df-0r 7682 df-1r 7683 df-1 7771 df-r 7773 |
This theorem is referenced by: nnindnn 7844 |
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