prproropf1olem1 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem1 44955

Description: Lemma 1 for prproropf1o 44959. (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 44954	. . 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 1194	. . . . 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 5363	. . . . 5 ⊢ (((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉)
5	3, 4	syl 17	. . . 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 5522	. . . . . . 7 ⊢ (𝑅 Or 𝑉 → 𝑅 Po 𝑉)
7	6	adantr 481	. . . . . 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 1195	. . . . . 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 5519	. . . . . 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 1370	. . . . 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 6787	. . . . . 6 ⊢ (1^st ‘𝑊) ∈ V
12		fvex 6787	. . . . . 6 ⊢ (2^nd ‘𝑊) ∈ V
13		hashprg 14110	. . . . . 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 689	. . . . 5 ⊢ ((1^st ‘𝑊) ≠ (2^nd ‘𝑊) ↔ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2)
15	10, 14	sylib 217	. . . 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 512	. . 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 594	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → ({(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1^st ‘𝑊), (2^nd ‘𝑊)}) = 2))
18		fveqeq2 6783	. . 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 3627	. 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 233	1 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1^st ‘𝑊), (2^nd ‘𝑊)} ∈ 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 {crab 3068 Vcvv 3432 ∩ cin 3886 𝒫 cpw 4533 {cpr 4563 ⟨cop 4567 class class class wbr 5074 Po wpo 5501 Or wor 5502 × cxp 5587 ‘cfv 6433 1^st c1st 7829 2^nd c2nd 7830 2c2 12028 ♯chash 14044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-hash 14045

This theorem is referenced by: prproropf1olem3 44957 prproropf1o 44959

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