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| Mirrors > Home > MPE Home > Th. List > dflt2 | Structured version Visualization version GIF version | ||
| Description: Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.) |
| Ref | Expression |
|---|---|
| dflt2 | ⊢ < = ( ≤ ∖ I ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltrel 11323 | . 2 ⊢ Rel < | |
| 2 | difss 4136 | . . 3 ⊢ ( ≤ ∖ I ) ⊆ ≤ | |
| 3 | lerel 11325 | . . 3 ⊢ Rel ≤ | |
| 4 | relss 5791 | . . 3 ⊢ (( ≤ ∖ I ) ⊆ ≤ → (Rel ≤ → Rel ( ≤ ∖ I ))) | |
| 5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≤ ∖ I ) |
| 6 | ltrelxr 11322 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
| 7 | 6 | brel 5750 | . . 3 ⊢ (𝑥 < 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
| 8 | lerelxr 11324 | . . . . 5 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
| 9 | 2, 8 | sstri 3993 | . . . 4 ⊢ ( ≤ ∖ I ) ⊆ (ℝ* × ℝ*) |
| 10 | 9 | brel 5750 | . . 3 ⊢ (𝑥( ≤ ∖ I )𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
| 11 | xrltlen 13188 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥))) | |
| 12 | equcom 2017 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
| 13 | vex 3484 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
| 14 | 13 | ideq 5863 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
| 15 | 12, 14 | bitr4i 278 | . . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 I 𝑦) |
| 16 | 15 | necon3abii 2987 | . . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 I 𝑦) |
| 17 | 16 | anbi2i 623 | . . . . 5 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) |
| 18 | 11, 17 | bitrdi 287 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦))) |
| 19 | brdif 5196 | . . . 4 ⊢ (𝑥( ≤ ∖ I )𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) | |
| 20 | 18, 19 | bitr4di 289 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦)) |
| 21 | 7, 10, 20 | pm5.21nii 378 | . 2 ⊢ (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦) |
| 22 | 1, 5, 21 | eqbrriv 5801 | 1 ⊢ < = ( ≤ ∖ I ) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 I cid 5577 × cxp 5683 Rel wrel 5690 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: relt 21633 xrslt 33009 |
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