Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10860 | . 2 ⊢ Rel < | |
2 | difss 4032 | . . 3 ⊢ ( ≤ ∖ I ) ⊆ ≤ | |
3 | lerel 10862 | . . 3 ⊢ Rel ≤ | |
4 | relss 5638 | . . 3 ⊢ (( ≤ ∖ I ) ⊆ ≤ → (Rel ≤ → Rel ( ≤ ∖ I ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≤ ∖ I ) |
6 | ltrelxr 10859 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
7 | 6 | brel 5599 | . . 3 ⊢ (𝑥 < 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
8 | lerelxr 10861 | . . . . 5 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 2, 8 | sstri 3896 | . . . 4 ⊢ ( ≤ ∖ I ) ⊆ (ℝ* × ℝ*) |
10 | 9 | brel 5599 | . . 3 ⊢ (𝑥( ≤ ∖ I )𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrltlen 12701 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥))) | |
12 | equcom 2028 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
13 | vex 3402 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
14 | 13 | ideq 5706 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
15 | 12, 14 | bitr4i 281 | . . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 I 𝑦) |
16 | 15 | necon3abii 2978 | . . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 I 𝑦) |
17 | 16 | anbi2i 626 | . . . . 5 ⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) |
18 | 11, 17 | bitrdi 290 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦))) |
19 | brdif 5092 | . . . 4 ⊢ (𝑥( ≤ ∖ I )𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 I 𝑦)) | |
20 | 18, 19 | bitr4di 292 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦)) |
21 | 7, 10, 20 | pm5.21nii 383 | . 2 ⊢ (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦) |
22 | 1, 5, 21 | eqbrriv 5646 | 1 ⊢ < = ( ≤ ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 ⊆ wss 3853 class class class wbr 5039 I cid 5439 × cxp 5534 Rel wrel 5541 ℝ*cxr 10831 < clt 10832 ≤ cle 10833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-pre-lttri 10768 ax-pre-lttrn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 |
This theorem is referenced by: relt 20531 xrslt 30958 |
Copyright terms: Public domain | W3C validator |