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Mirrors > Home > ILE Home > Th. List > axpre-ltirr | GIF version |
Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 7700. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-ltirr | ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7604 | . . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | df-rex 2399 | . . 3 ⊢ (∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ↔ ∃𝑥(𝑥 ∈ R ∧ 〈𝑥, 0R〉 = 𝐴)) | |
3 | 1, 2 | bitri 183 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥(𝑥 ∈ R ∧ 〈𝑥, 0R〉 = 𝐴)) |
4 | id 19 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → 〈𝑥, 0R〉 = 𝐴) | |
5 | 4, 4 | breq12d 3912 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝐴 <ℝ 𝐴)) |
6 | 5 | notbid 641 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → (¬ 〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ ¬ 𝐴 <ℝ 𝐴)) |
7 | ltsosr 7540 | . . . . 5 ⊢ <R Or R | |
8 | ltrelsr 7514 | . . . . 5 ⊢ <R ⊆ (R × R) | |
9 | 7, 8 | soirri 4903 | . . . 4 ⊢ ¬ 𝑥 <R 𝑥 |
10 | ltresr 7615 | . . . 4 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑥 <R 𝑥) | |
11 | 9, 10 | mtbir 645 | . . 3 ⊢ ¬ 〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 |
12 | 11 | a1i 9 | . 2 ⊢ (𝑥 ∈ R → ¬ 〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉) |
13 | 3, 6, 12 | gencl 2692 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ∃wrex 2394 〈cop 3500 class class class wbr 3899 Rcnr 7073 0Rc0r 7074 <R cltr 7079 ℝcr 7587 <ℝ cltrr 7592 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-i1p 7243 df-iplp 7244 df-iltp 7246 df-enr 7502 df-nr 7503 df-ltr 7506 df-0r 7507 df-r 7598 df-lt 7601 |
This theorem is referenced by: (None) |
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