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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropf1olem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for prproropf1o 43676. (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 43671 | . . 3 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
3 | simpr2 1191 | . . . . 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 5343 | . . . . 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 5495 | . . . . . . 7 ⊢ (𝑅 Or 𝑉 → 𝑅 Po 𝑉) | |
7 | 6 | adantr 483 | . . . . . 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 1192 | . . . . . 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 5492 | . . . . . 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 1367 | . . . . 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 6686 | . . . . . 6 ⊢ (1st ‘𝑊) ∈ V | |
12 | fvex 6686 | . . . . . 6 ⊢ (2nd ‘𝑊) ∈ V | |
13 | hashprg 13759 | . . . . . 6 ⊢ (((1st ‘𝑊) ∈ V ∧ (2nd ‘𝑊) ∈ V) → ((1st ‘𝑊) ≠ (2nd ‘𝑊) ↔ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) | |
14 | 11, 12, 13 | mp2an 690 | . . . . 5 ⊢ ((1st ‘𝑊) ≠ (2nd ‘𝑊) ↔ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2) |
15 | 10, 14 | sylib 220 | . . . 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 514 | . . 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 6682 | . . 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 3686 | . 2 ⊢ ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃 ↔ ({(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 2)) |
21 | 17, 20 | sylibr 236 | 1 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 {crab 3145 Vcvv 3497 ∩ cin 3938 𝒫 cpw 4542 {cpr 4572 〈cop 4576 class class class wbr 5069 Po wpo 5475 Or wor 5476 × cxp 5556 ‘cfv 6358 1st c1st 7690 2nd c2nd 7691 2c2 11695 ♯chash 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: prproropf1olem3 43674 prproropf1o 43676 |
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