prproropf1olem1 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem1 48280

Description: Lemma 1 for prproropf1o 48284. (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 48279	. . 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 5428	. . . . 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 5588	. . . . . . 7 ⊢ (𝑅 Or 𝑉 → 𝑅 Po 𝑉)
7	6	adantr 485	. . . . . 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 5585	. . . . . 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 6894	. . . . . 6 ⊢ (1^st ‘𝑊) ∈ V
12		fvex 6894	. . . . . 6 ⊢ (2^nd ‘𝑊) ∈ V
13		hashprg 14436	. . . . . 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 704	. . . . 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 520	. . 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 605	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → ({(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2))
18		fveqeq2 6890	. . 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 3654	. 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 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 {crab 3416 Vcvv 3455 ∩ cin 3904 𝒫 cpw 4562 {cpr 4591 ⟨cop 4595 class class class wbr 5109 Po wpo 5567 Or wor 5568 × cxp 5659 ‘cfv 6536 1^st c1st 7980 2^nd c2nd 7981 2c2 12299 ♯chash 14371

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9892 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-n0 12509 df-z 12596 df-uz 12867 df-fz 13540 df-hash 14372

This theorem is used by: prproropf1olem3 48282 prproropf1o 48284

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