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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 10550 | . 2 ⊢ Rel < | |
2 | difss 4029 | . . 3 ⊢ ( ≤ ∖ I ) ⊆ ≤ | |
3 | lerel 10552 | . . 3 ⊢ Rel ≤ | |
4 | relss 5542 | . . 3 ⊢ (( ≤ ∖ I ) ⊆ ≤ → (Rel ≤ → Rel ( ≤ ∖ I ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≤ ∖ I ) |
6 | ltrelxr 10549 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
7 | 6 | brel 5503 | . . 3 ⊢ (𝑥 < 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
8 | lerelxr 10551 | . . . . 5 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 2, 8 | sstri 3898 | . . . 4 ⊢ ( ≤ ∖ I ) ⊆ (ℝ* × ℝ*) |
10 | 9 | brel 5503 | . . 3 ⊢ (𝑥( ≤ ∖ I )𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrltlen 12389 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥))) | |
12 | equcom 2002 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
13 | vex 3440 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
14 | 13 | ideq 5609 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
15 | 12, 14 | bitr4i 279 | . . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 I 𝑦) |
16 | 15 | necon3abii 3030 | . . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 I 𝑦) |
17 | 16 | anbi2i 622 | . . . . 5 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) |
18 | 11, 17 | syl6bb 288 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦))) |
19 | brdif 5015 | . . . 4 ⊢ (𝑥( ≤ ∖ I )𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) | |
20 | 18, 19 | syl6bbr 290 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦)) |
21 | 7, 10, 20 | pm5.21nii 380 | . 2 ⊢ (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦) |
22 | 1, 5, 21 | eqbrriv 5550 | 1 ⊢ < = ( ≤ ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∖ cdif 3856 ⊆ wss 3859 class class class wbr 4962 I cid 5347 × cxp 5441 Rel wrel 5448 ℝ*cxr 10520 < clt 10521 ≤ cle 10522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-pre-lttri 10457 ax-pre-lttrn 10458 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 |
This theorem is referenced by: relt 20441 xrslt 30337 |
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