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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropf1olem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for prproropf1o 47431. (Contributed by AV, 12-Mar-2023.) |
Ref | Expression |
---|---|
prproropf1o.o | ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) |
prproropf1o.p | ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} |
Ref | Expression |
---|---|
prproropf1olem1 | ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prproropf1o.o | . . . 4 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) | |
2 | 1 | prproropf1olem0 47426 | . . 3 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
3 | simpr2 1194 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) → ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)) | |
4 | prelpwi 5457 | . . . . 5 ⊢ (((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉) |
6 | sopo 5615 | . . . . . . 7 ⊢ (𝑅 Or 𝑉 → 𝑅 Po 𝑉) | |
7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) → 𝑅 Po 𝑉) |
8 | simpr3 1195 | . . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) → (1st ‘𝑊)𝑅(2nd ‘𝑊)) | |
9 | po2ne 5612 | . . . . . 6 ⊢ ((𝑅 Po 𝑉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊)) → (1st ‘𝑊) ≠ (2nd ‘𝑊)) | |
10 | 7, 3, 8, 9 | syl3anc 1370 | . . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) → (1st ‘𝑊) ≠ (2nd ‘𝑊)) |
11 | fvex 6919 | . . . . . 6 ⊢ (1st ‘𝑊) ∈ V | |
12 | fvex 6919 | . . . . . 6 ⊢ (2nd ‘𝑊) ∈ V | |
13 | hashprg 14430 | . . . . . 6 ⊢ (((1st ‘𝑊) ∈ V ∧ (2nd ‘𝑊) ∈ V) → ((1st ‘𝑊) ≠ (2nd ‘𝑊) ↔ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) | |
14 | 11, 12, 13 | mp2an 692 | . . . . 5 ⊢ ((1st ‘𝑊) ≠ (2nd ‘𝑊) ↔ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2) |
15 | 10, 14 | sylib 218 | . . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) → (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2) |
16 | 5, 15 | jca 511 | . . 3 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) → ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) |
17 | 2, 16 | sylan2b 594 | . 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) |
18 | fveqeq2 6915 | . . 3 ⊢ (𝑝 = {(1st ‘𝑊), (2nd ‘𝑊)} → ((♯‘𝑝) = 2 ↔ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) | |
19 | prproropf1o.p | . . 3 ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} | |
20 | 18, 19 | elrab2 3697 | . 2 ⊢ ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃 ↔ ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) |
21 | 17, 20 | sylibr 234 | 1 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 {crab 3432 Vcvv 3477 ∩ cin 3961 𝒫 cpw 4604 {cpr 4632 〈cop 4636 class class class wbr 5147 Po wpo 5594 Or wor 5595 × cxp 5686 ‘cfv 6562 1st c1st 8010 2nd c2nd 8011 2c2 12318 ♯chash 14365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366 |
This theorem is referenced by: prproropf1olem3 47429 prproropf1o 47431 |
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