prproropf1olem1 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem1 48501

Description: Lemma 1 for prproropf1o 48505. (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 48500	. . 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 5415	. . . . 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 5575	. . . . . . 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 5572	. . . . . 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 6887	. . . . . 6 ⊢ (1^st ‘𝑊) ∈ V
12		fvex 6887	. . . . . 6 ⊢ (2^nd ‘𝑊) ∈ V
13		hashprg 14492	. . . . . 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 6883	. . 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 3649	. 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 1570 ∈ wcel 2145 ≠ wne 2955 {crab 3412 Vcvv 3450 ∩ cin 3898 𝒫 cpw 4557 {cpr 4586 ⟨cop 4590 class class class wbr 5103 Po wpo 5554 Or wor 5555 × cxp 5646 ‘cfv 6528 1^st c1st 7983 2^nd c2nd 7984 2c2 12352 ♯chash 14427

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-oadd 8459 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9939 df-card 9977 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-nn 12291 df-2 12360 df-n0 12562 df-z 12649 df-uz 12921 df-fz 13595 df-hash 14428

This theorem is used by: prproropf1olem3 48503 prproropf1o 48505

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