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| 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 11326 | . 2 ⊢ Rel ≤ | |
| 2 | ltrelxr 11323 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
| 3 | idssxp 6066 | . . . 4 ⊢ ( I ↾ ℝ*) ⊆ (ℝ* × ℝ*) | |
| 4 | 2, 3 | unssi 4190 | . . 3 ⊢ ( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) | 
| 5 | relxp 5702 | . . 3 ⊢ Rel (ℝ* × ℝ*) | |
| 6 | relss 5790 | . . 3 ⊢ (( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ( < ∪ ( I ↾ ℝ*)))) | |
| 7 | 4, 5, 6 | mp2 9 | . 2 ⊢ Rel ( < ∪ ( I ↾ ℝ*)) | 
| 8 | lerelxr 11325 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
| 9 | 8 | brel 5749 | . . 3 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) | 
| 10 | 4 | brel 5749 | . . 3 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) | 
| 11 | xrleloe 13187 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) | |
| 12 | resieq 6007 | . . . . . 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 5193 | . . . 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 5800 | 1 ⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 class class class wbr 5142 I cid 5576 × cxp 5682 ↾ cres 5686 Rel wrel 5689 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 | 
| This theorem is referenced by: (None) | 
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