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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 7378. (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 7287 | . . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | df-rex 2361 | . . 3 ⊢ (∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ↔ ∃𝑥(𝑥 ∈ R ∧ 〈𝑥, 0R〉 = 𝐴)) | |
3 | 1, 2 | bitri 182 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥(𝑥 ∈ R ∧ 〈𝑥, 0R〉 = 𝐴)) |
4 | id 19 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → 〈𝑥, 0R〉 = 𝐴) | |
5 | 4, 4 | breq12d 3827 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝐴 <ℝ 𝐴)) |
6 | 5 | notbid 625 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → (¬ 〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ ¬ 𝐴 <ℝ 𝐴)) |
7 | ltsosr 7231 | . . . . 5 ⊢ <R Or R | |
8 | ltrelsr 7205 | . . . . 5 ⊢ <R ⊆ (R × R) | |
9 | 7, 8 | soirri 4784 | . . . 4 ⊢ ¬ 𝑥 <R 𝑥 |
10 | ltresr 7297 | . . . 4 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑥 <R 𝑥) | |
11 | 9, 10 | mtbir 629 | . . 3 ⊢ ¬ 〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉 |
12 | 11 | a1i 9 | . 2 ⊢ (𝑥 ∈ R → ¬ 〈𝑥, 0R〉 <ℝ 〈𝑥, 0R〉) |
13 | 3, 6, 12 | gencl 2644 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1287 ∃wex 1424 ∈ wcel 1436 ∃wrex 2356 〈cop 3428 class class class wbr 3814 Rcnr 6777 0Rc0r 6778 <R cltr 6783 ℝcr 7270 <ℝ cltrr 7275 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 |
This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-eprel 4083 df-id 4087 df-po 4090 df-iso 4091 df-iord 4160 df-on 4162 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-recs 6005 df-irdg 6070 df-1o 6116 df-2o 6117 df-oadd 6120 df-omul 6121 df-er 6225 df-ec 6227 df-qs 6231 df-ni 6784 df-pli 6785 df-mi 6786 df-lti 6787 df-plpq 6824 df-mpq 6825 df-enq 6827 df-nqqs 6828 df-plqqs 6829 df-mqqs 6830 df-1nqqs 6831 df-rq 6832 df-ltnqqs 6833 df-enq0 6904 df-nq0 6905 df-0nq0 6906 df-plq0 6907 df-mq0 6908 df-inp 6946 df-i1p 6947 df-iplp 6948 df-iltp 6950 df-enr 7193 df-nr 7194 df-ltr 7197 df-0r 7198 df-r 7281 df-lt 7284 |
This theorem is referenced by: (None) |
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