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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropf1olem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for prproropf1o 43739. (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 43734 | . . 3 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
3 | simpr2 1190 | . . . . 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 5333 | . . . . 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 5485 | . . . . . . 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 1191 | . . . . . 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 5482 | . . . . . 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 1366 | . . . . 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 6676 | . . . . . 6 ⊢ (1st ‘𝑊) ∈ V | |
12 | fvex 6676 | . . . . . 6 ⊢ (2nd ‘𝑊) ∈ V | |
13 | hashprg 13753 | . . . . . 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 6672 | . . 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 3679 | . 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 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 {crab 3141 Vcvv 3491 ∩ cin 3928 𝒫 cpw 4532 {cpr 4562 〈cop 4566 class class class wbr 5059 Po wpo 5465 Or wor 5466 × cxp 5546 ‘cfv 6348 1st c1st 7680 2nd c2nd 7681 2c2 11686 ♯chash 13687 |
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 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-hash 13688 |
This theorem is referenced by: prproropf1olem3 43737 prproropf1o 43739 |
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