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Mirrors > Home > MPE Home > Th. List > ipole | Structured version Visualization version GIF version |
Description: Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
ipoval.i | ⊢ 𝐼 = (toInc‘𝐹) |
ipole.l | ⊢ ≤ = (le‘𝐼) |
Ref | Expression |
---|---|
ipole | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 ≤ 𝑌 ↔ 𝑋 ⊆ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12 4502 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌}) | |
2 | 1 | sseq1d 3851 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥, 𝑦} ⊆ 𝐹 ↔ {𝑋, 𝑌} ⊆ 𝐹)) |
3 | sseq12 3847 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑌)) | |
4 | 2, 3 | anbi12d 624 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦) ↔ ({𝑋, 𝑌} ⊆ 𝐹 ∧ 𝑋 ⊆ 𝑌))) |
5 | eqid 2778 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} | |
6 | 4, 5 | brabga 5226 | . . 3 ⊢ ((𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}𝑌 ↔ ({𝑋, 𝑌} ⊆ 𝐹 ∧ 𝑋 ⊆ 𝑌))) |
7 | 6 | 3adant1 1121 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}𝑌 ↔ ({𝑋, 𝑌} ⊆ 𝐹 ∧ 𝑋 ⊆ 𝑌))) |
8 | ipoval.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐹) | |
9 | 8 | ipolerval 17542 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) |
10 | ipole.l | . . . . 5 ⊢ ≤ = (le‘𝐼) | |
11 | 9, 10 | syl6reqr 2833 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}) |
12 | 11 | breqd 4897 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑋 ≤ 𝑌 ↔ 𝑋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}𝑌)) |
13 | 12 | 3ad2ant1 1124 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 ≤ 𝑌 ↔ 𝑋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}𝑌)) |
14 | prssi 4583 | . . . 4 ⊢ ((𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → {𝑋, 𝑌} ⊆ 𝐹) | |
15 | 14 | 3adant1 1121 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → {𝑋, 𝑌} ⊆ 𝐹) |
16 | 15 | biantrurd 528 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 ⊆ 𝑌 ↔ ({𝑋, 𝑌} ⊆ 𝐹 ∧ 𝑋 ⊆ 𝑌))) |
17 | 7, 13, 16 | 3bitr4d 303 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 ≤ 𝑌 ↔ 𝑋 ⊆ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 {cpr 4400 class class class wbr 4886 {copab 4948 ‘cfv 6135 lecple 16345 toInccipo 17537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-tset 16357 df-ple 16358 df-ocomp 16359 df-ipo 17538 |
This theorem is referenced by: ipolt 17545 ipopos 17546 isipodrs 17547 ipodrsfi 17549 mrelatglb 17570 mrelatglb0 17571 mrelatlub 17572 thlleval 20441 |
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