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Mirrors > Home > ILE Home > Th. List > ax1re | GIF version |
Description: 1 is a real number.
Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-1re 7904.
In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7903 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax1re | ⊢ 1 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1 7818 | . 2 ⊢ 1 = 〈1R, 0R〉 | |
2 | 1sr 7749 | . . 3 ⊢ 1R ∈ R | |
3 | opelreal 7825 | . . 3 ⊢ (〈1R, 0R〉 ∈ ℝ ↔ 1R ∈ R) | |
4 | 2, 3 | mpbir 146 | . 2 ⊢ 〈1R, 0R〉 ∈ ℝ |
5 | 1, 4 | eqeltri 2250 | 1 ⊢ 1 ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 〈cop 3595 Rcnr 7295 0Rc0r 7296 1Rc1r 7297 ℝcr 7809 1c1 7811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-eprel 4289 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-1o 6416 df-2o 6417 df-oadd 6420 df-omul 6421 df-er 6534 df-ec 6536 df-qs 6540 df-ni 7302 df-pli 7303 df-mi 7304 df-lti 7305 df-plpq 7342 df-mpq 7343 df-enq 7345 df-nqqs 7346 df-plqqs 7347 df-mqqs 7348 df-1nqqs 7349 df-rq 7350 df-ltnqqs 7351 df-enq0 7422 df-nq0 7423 df-0nq0 7424 df-plq0 7425 df-mq0 7426 df-inp 7464 df-i1p 7465 df-iplp 7466 df-enr 7724 df-nr 7725 df-0r 7729 df-1r 7730 df-1 7818 df-r 7820 |
This theorem is referenced by: peano5nnnn 7890 |
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