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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 11220 | . 2 ⊢ Rel ≤ | |
2 | ltrelxr 11217 | . . . 4 ⊢ < ⊆ (ℝ* × ℝ*) | |
3 | idssxp 6003 | . . . 4 ⊢ ( I ↾ ℝ*) ⊆ (ℝ* × ℝ*) | |
4 | 2, 3 | unssi 4146 | . . 3 ⊢ ( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) |
5 | relxp 5652 | . . 3 ⊢ Rel (ℝ* × ℝ*) | |
6 | relss 5738 | . . 3 ⊢ (( < ∪ ( I ↾ ℝ*)) ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ( < ∪ ( I ↾ ℝ*)))) | |
7 | 4, 5, 6 | mp2 9 | . 2 ⊢ Rel ( < ∪ ( I ↾ ℝ*)) |
8 | lerelxr 11219 | . . . 4 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
9 | 8 | brel 5698 | . . 3 ⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
10 | 4 | brel 5698 | . . 3 ⊢ (𝑥( < ∪ ( I ↾ ℝ*))𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) |
11 | xrleloe 13064 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))) | |
12 | resieq 5949 | . . . . . 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 5157 | . . . 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 380 | . 2 ⊢ (𝑥 ≤ 𝑦 ↔ 𝑥( < ∪ ( I ↾ ℝ*))𝑦) |
18 | 1, 7, 17 | eqbrriv 5748 | 1 ⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 ⊆ wss 3911 class class class wbr 5106 I cid 5531 × cxp 5632 ↾ cres 5636 Rel wrel 5639 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 |
This theorem is referenced by: (None) |
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