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Mirrors > Home > MPE Home > Th. List > dfle2 | Structured version Visualization version GIF version |
Description: Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.) |
Ref | Expression |
---|---|
dfle2 | ⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lerel 11323 | . 2 ⊢ Rel ≤ | |
2 | ltrelxr 11320 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
3 | idssxp 6069 | . . . 4 ⊢ ( I ↾ ℝ*) ⊆ (ℝ* × ℝ*) | |
4 | 2, 3 | unssi 4201 | . . 3 ⊢ ( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) |
5 | relxp 5707 | . . 3 ⊢ Rel (ℝ* × ℝ*) | |
6 | relss 5794 | . . 3 ⊢ (( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ( < ∪ ( I ↾ ℝ*)))) | |
7 | 4, 5, 6 | mp2 9 | . 2 ⊢ Rel ( < ∪ ( I ↾ ℝ*)) |
8 | lerelxr 11322 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 8 | brel 5754 | . . 3 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
10 | 4 | brel 5754 | . . 3 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrleloe 13183 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) | |
12 | resieq 6011 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥( I ↾ ℝ*)𝑦 ↔ 𝑥 = 𝑦)) | |
13 | 12 | orbi2d 915 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → ((𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦) ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) |
14 | 11, 13 | bitr4d 282 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦))) |
15 | brun 5199 | . . . 4 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦)) | |
16 | 14, 15 | bitr4di 289 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ 𝑥( < ∪ ( I ↾ ℝ*))𝑦)) |
17 | 9, 10, 16 | pm5.21nii 378 | . 2 ⊢ (𝑥 ≤ 𝑦 ↔ 𝑥( < ∪ ( I ↾ ℝ*))𝑦) |
18 | 1, 7, 17 | eqbrriv 5804 | 1 ⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 class class class wbr 5148 I cid 5582 × cxp 5687 ↾ cres 5691 Rel wrel 5694 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 |
This theorem is referenced by: (None) |
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