prproropf1olem1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropf1olem1		Structured version Visualization version GIF version

Theorem prproropf1olem1 48585

Description: Lemma 1 for prproropf1o 48589. (Contributed by AV, 12-Mar-2023.)

**Hypotheses**
Ref	Expression
prproropf1o.o	⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
prproropf1o.p	⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}

**Assertion**
Ref	Expression
prproropf1olem1	⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)

Distinct variable groups: 𝑉,𝑝 𝑊,𝑝 Allowed substitution hints: 𝑃(𝑝) 𝑅(𝑝) 𝑂(𝑝)

**Proof of Theorem prproropf1olem1**
Step	Hyp	Ref	Expression
1		prproropf1o.o	. . . 4 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
2	1	prproropf1olem0 48584	. . 3 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
3		simpr2 1214	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉))
4		prelpwi 5414	. . . . 5 ⊢ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉)
5	3, 4	syl 18	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉)
6		sopo 5577	. . . . . . 7 ⊢ (𝑅 Or 𝑉 → 𝑅 Po 𝑉)
7	6	adantr 486	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → 𝑅 Po 𝑉)
8		simpr3 1215	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → (1^st ‘𝑊)𝑅(2^nd ‘𝑊))
9		po2ne 5574	. . . . . 6 ⊢ ((𝑅 Po 𝑉 ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)) → (1^st ‘𝑊) ≠ (2^nd ‘𝑊))
10	7, 3, 8, 9	syl3anc 1398	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → (1^st ‘𝑊) ≠ (2^nd ‘𝑊))
11		fvex 6898	. . . . . 6 ⊢ (1^st ‘𝑊) ∈ V
12		fvex 6898	. . . . . 6 ⊢ (2^nd ‘𝑊) ∈ V
13		hashprg 14539	. . . . . 6 ⊢ (((1^st ‘𝑊) ∈ V ∧ (2^nd ‘𝑊) ∈ V) → ((1^st ‘𝑊) ≠ (2^nd ‘𝑊) ↔ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2))
14	11, 12, 13	mp2an 705	. . . . 5 ⊢ ((1^st ‘𝑊) ≠ (2^nd ‘𝑊) ↔ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2)
15	10, 14	sylib 221	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2)
16	5, 15	jca 521	. . 3 ⊢ ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊))) → ({(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2))
17	2, 16	sylan2b 606	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → ({(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2))
18		fveqeq2 6894	. . 3 ⊢ (𝑝 = {(1^st ‘𝑊), (2^nd ‘𝑊)} → ((♯‘𝑝) = 2 ↔ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2))
19		prproropf1o.p	. . 3 ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
20	18, 19	elrab2 3648	. 2 ⊢ ({(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃 ↔ ({(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2))
21	17, 20	sylibr 237	1 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 {crab 3412 Vcvv 3450 ∩ cin 4089 𝒫 cpw 4556 {cpr 4585 ⟨cop 4589 class class class wbr 5102 Po wpo 5556 Or wor 5557 × cxp 5648 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000 2c2 12397 ♯chash 14474

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-oadd 8480 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-hash 14475

This theorem is used by: prproropf1olem3 48587 prproropf1o 48589

W3C validator