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Mirrors > Home > ILE Home > Th. List > gtso | GIF version |
Description: 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
Ref | Expression |
---|---|
gtso | ⊢ ◡ < Or ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 7835 | . 2 ⊢ < Or ℝ | |
2 | 0re 7759 | . . 3 ⊢ 0 ∈ ℝ | |
3 | elex2 2697 | . . 3 ⊢ (0 ∈ ℝ → ∃𝑥 𝑥 ∈ ℝ) | |
4 | cnvsom 5077 | . . 3 ⊢ (∃𝑥 𝑥 ∈ ℝ → ( < Or ℝ ↔ ◡ < Or ℝ)) | |
5 | 2, 3, 4 | mp2b 8 | . 2 ⊢ ( < Or ℝ ↔ ◡ < Or ℝ) |
6 | 1, 5 | mpbi 144 | 1 ⊢ ◡ < Or ℝ |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1468 ∈ wcel 1480 Or wor 4212 ◡ccnv 4533 ℝcr 7612 0cc0 7613 < clt 7793 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-po 4213 df-iso 4214 df-xp 4540 df-cnv 4542 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: (None) |
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