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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 11308 | . 2 ⊢ Rel ≤ | |
2 | ltrelxr 11305 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
3 | idssxp 6052 | . . . 4 ⊢ ( I ↾ ℝ*) ⊆ (ℝ* × ℝ*) | |
4 | 2, 3 | unssi 4185 | . . 3 ⊢ ( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) |
5 | relxp 5696 | . . 3 ⊢ Rel (ℝ* × ℝ*) | |
6 | relss 5783 | . . 3 ⊢ (( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ( < ∪ ( I ↾ ℝ*)))) | |
7 | 4, 5, 6 | mp2 9 | . 2 ⊢ Rel ( < ∪ ( I ↾ ℝ*)) |
8 | lerelxr 11307 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 8 | brel 5743 | . . 3 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
10 | 4 | brel 5743 | . . 3 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrleloe 13155 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) | |
12 | resieq 5996 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥( I ↾ ℝ*)𝑦 ↔ 𝑥 = 𝑦)) | |
13 | 12 | orbi2d 914 | . . . . 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 5793 | 1 ⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 ⊆ wss 3947 class class class wbr 5148 I cid 5575 × cxp 5676 ↾ cres 5680 Rel wrel 5683 ℝ*cxr 11277 < clt 11278 ≤ cle 11279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 |
This theorem is referenced by: (None) |
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