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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 11218 | . 2 ⊢ Rel < | |
2 | difss 4092 | . . 3 ⊢ ( ≤ ∖ I ) ⊆ ≤ | |
3 | lerel 11220 | . . 3 ⊢ Rel ≤ | |
4 | relss 5738 | . . 3 ⊢ (( ≤ ∖ I ) ⊆ ≤ → (Rel ≤ → Rel ( ≤ ∖ I ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≤ ∖ I ) |
6 | ltrelxr 11217 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
7 | 6 | brel 5698 | . . 3 ⊢ (𝑥 < 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
8 | lerelxr 11219 | . . . . 5 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 2, 8 | sstri 3954 | . . . 4 ⊢ ( ≤ ∖ I ) ⊆ (ℝ* × ℝ*) |
10 | 9 | brel 5698 | . . 3 ⊢ (𝑥( ≤ ∖ I )𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrltlen 13066 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥))) | |
12 | equcom 2022 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
13 | vex 3450 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
14 | 13 | ideq 5809 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
15 | 12, 14 | bitr4i 278 | . . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 I 𝑦) |
16 | 15 | necon3abii 2991 | . . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 I 𝑦) |
17 | 16 | anbi2i 624 | . . . . 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 5159 | . . . 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 380 | . 2 ⊢ (𝑥 < 𝑦 ↔ 𝑥( ≤ ∖ I )𝑦) |
22 | 1, 5, 21 | eqbrriv 5748 | 1 ⊢ < = ( ≤ ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∖ cdif 3908 ⊆ wss 3911 class class class wbr 5106 I cid 5531 × cxp 5632 Rel wrel 5639 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 |
This theorem is referenced by: relt 21022 xrslt 31870 |
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