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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropf1olem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for prproropf1o 45773. (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 45768 | . . 3 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
3 | simpr2 1196 | . . . . 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 5409 | . . . . 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 5569 | . . . . . . 7 ⊢ (𝑅 Or 𝑉 → 𝑅 Po 𝑉) | |
7 | 6 | adantr 482 | . . . . . 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 1197 | . . . . . 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 5566 | . . . . . 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 1372 | . . . . 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 6860 | . . . . . 6 ⊢ (1st ‘𝑊) ∈ V | |
12 | fvex 6860 | . . . . . 6 ⊢ (2nd ‘𝑊) ∈ V | |
13 | hashprg 14302 | . . . . . 6 ⊢ (((1st ‘𝑊) ∈ V ∧ (2nd ‘𝑊) ∈ V) → ((1st ‘𝑊) ≠ (2nd ‘𝑊) ↔ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) | |
14 | 11, 12, 13 | mp2an 691 | . . . . 5 ⊢ ((1st ‘𝑊) ≠ (2nd ‘𝑊) ↔ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2) |
15 | 10, 14 | sylib 217 | . . . 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 513 | . . 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 595 | . 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) |
18 | fveqeq2 6856 | . . 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 3653 | . 2 ⊢ ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃 ↔ ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) |
21 | 17, 20 | sylibr 233 | 1 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 {crab 3410 Vcvv 3448 ∩ cin 3914 𝒫 cpw 4565 {cpr 4593 ⟨cop 4597 class class class wbr 5110 Po wpo 5548 Or wor 5549 × cxp 5636 ‘cfv 6501 1st c1st 7924 2nd c2nd 7925 2c2 12215 ♯chash 14237 |
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 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: prproropf1olem3 45771 prproropf1o 45773 |
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