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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 10702 | . 2 ⊢ Rel < | |
2 | difss 4107 | . . 3 ⊢ ( ≤ ∖ I ) ⊆ ≤ | |
3 | lerel 10704 | . . 3 ⊢ Rel ≤ | |
4 | relss 5655 | . . 3 ⊢ (( ≤ ∖ I ) ⊆ ≤ → (Rel ≤ → Rel ( ≤ ∖ I ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≤ ∖ I ) |
6 | ltrelxr 10701 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
7 | 6 | brel 5616 | . . 3 ⊢ (𝑥 < 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
8 | lerelxr 10703 | . . . . 5 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 2, 8 | sstri 3975 | . . . 4 ⊢ ( ≤ ∖ I ) ⊆ (ℝ* × ℝ*) |
10 | 9 | brel 5616 | . . 3 ⊢ (𝑥( ≤ ∖ I )𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrltlen 12538 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥))) | |
12 | equcom 2021 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
13 | vex 3497 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
14 | 13 | ideq 5722 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
15 | 12, 14 | bitr4i 280 | . . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 I 𝑦) |
16 | 15 | necon3abii 3062 | . . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 I 𝑦) |
17 | 16 | anbi2i 624 | . . . . 5 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) |
18 | 11, 17 | syl6bb 289 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦))) |
19 | brdif 5118 | . . . 4 ⊢ (𝑥( ≤ ∖ I )𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) | |
20 | 18, 19 | syl6bbr 291 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦)) |
21 | 7, 10, 20 | pm5.21nii 382 | . 2 ⊢ (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦) |
22 | 1, 5, 21 | eqbrriv 5663 | 1 ⊢ < = ( ≤ ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 ⊆ wss 3935 class class class wbr 5065 I cid 5458 × cxp 5552 Rel wrel 5559 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-pre-lttri 10610 ax-pre-lttrn 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 |
This theorem is referenced by: relt 20758 xrslt 30663 |
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