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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 10699 | . 2 ⊢ Rel ≤ | |
2 | ltrelxr 10696 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
3 | idssxp 5910 | . . . 4 ⊢ ( I ↾ ℝ*) ⊆ (ℝ* × ℝ*) | |
4 | 2, 3 | unssi 4160 | . . 3 ⊢ ( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) |
5 | relxp 5567 | . . 3 ⊢ Rel (ℝ* × ℝ*) | |
6 | relss 5650 | . . 3 ⊢ (( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ( < ∪ ( I ↾ ℝ*)))) | |
7 | 4, 5, 6 | mp2 9 | . 2 ⊢ Rel ( < ∪ ( I ↾ ℝ*)) |
8 | lerelxr 10698 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 8 | brel 5611 | . . 3 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
10 | 4 | brel 5611 | . . 3 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrleloe 12531 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) | |
12 | resieq 5858 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥( I ↾ ℝ*)𝑦 ↔ 𝑥 = 𝑦)) | |
13 | 12 | orbi2d 912 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → ((𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦) ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) |
14 | 11, 13 | bitr4d 284 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦))) |
15 | brun 5109 | . . . 4 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦)) | |
16 | 14, 15 | syl6bbr 291 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ 𝑥( < ∪ ( I ↾ ℝ*))𝑦)) |
17 | 9, 10, 16 | pm5.21nii 382 | . 2 ⊢ (𝑥 ≤ 𝑦 ↔ 𝑥( < ∪ ( I ↾ ℝ*))𝑦) |
18 | 1, 7, 17 | eqbrriv 5658 | 1 ⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 ⊆ wss 3935 class class class wbr 5058 I cid 5453 × cxp 5547 ↾ cres 5551 Rel wrel 5554 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 |
This theorem is referenced by: (None) |
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