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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 9981 | . 2 ⊢ Rel ≤ | |
2 | ltrelxr 9978 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
3 | f1oi 6086 | . . . . 5 ⊢ ( I ↾ ℝ*):ℝ*–1-1-onto→ℝ* | |
4 | f1of 6050 | . . . . 5 ⊢ (( I ↾ ℝ*):ℝ*–1-1-onto→ℝ* → ( I ↾ ℝ*):ℝ*⟶ℝ*) | |
5 | fssxp 5973 | . . . . 5 ⊢ (( I ↾ ℝ*):ℝ*⟶ℝ* → ( I ↾ ℝ*) ⊆ (ℝ* × ℝ*)) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ( I ↾ ℝ*) ⊆ (ℝ* × ℝ*) |
7 | 2, 6 | unssi 3750 | . . 3 ⊢ ( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) |
8 | relxp 5150 | . . 3 ⊢ Rel (ℝ* × ℝ*) | |
9 | relss 5129 | . . 3 ⊢ (( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ( < ∪ ( I ↾ ℝ*)))) | |
10 | 7, 8, 9 | mp2 9 | . 2 ⊢ Rel ( < ∪ ( I ↾ ℝ*)) |
11 | lerelxr 9980 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
12 | 11 | brel 5090 | . . 3 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
13 | 7 | brel 5090 | . . 3 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
14 | xrleloe 11853 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) | |
15 | resieq 5327 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥( I ↾ ℝ*)𝑦 ↔ 𝑥 = 𝑦)) | |
16 | 15 | orbi2d 734 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → ((𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦) ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) |
17 | 14, 16 | bitr4d 270 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦))) |
18 | brun 4633 | . . . 4 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥( I ↾ ℝ*)𝑦)) | |
19 | 17, 18 | syl6bbr 277 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ 𝑥( < ∪ ( I ↾ ℝ*))𝑦)) |
20 | 12, 13, 19 | pm5.21nii 367 | . 2 ⊢ (𝑥 ≤ 𝑦 ↔ 𝑥( < ∪ ( I ↾ ℝ*))𝑦) |
21 | 1, 10, 20 | eqbrriv 5138 | 1 ⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 class class class wbr 4583 I cid 4948 × cxp 5036 ↾ cres 5040 Rel wrel 5043 ⟶wf 5800 –1-1-onto→wf1o 5803 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: (None) |
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