prproropf1olem1 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem1 45685

Description: Lemma 1 for prproropf1o 45689. (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 45684	. . 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 1195	. . . . 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 5404	. . . . 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 5564	. . . . . . 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 1196	. . . . . 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 5561	. . . . . 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 1371	. . . . 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 6855	. . . . . 6 ⊢ (1^st ‘𝑊) ∈ V
12		fvex 6855	. . . . . 6 ⊢ (2^nd ‘𝑊) ∈ V
13		hashprg 14295	. . . . . 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 690	. . . . 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 6851	. . 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 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 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 {crab 3407 Vcvv 3445 ∩ cin 3909 𝒫 cpw 4560 {cpr 4588 ⟨cop 4592 class class class wbr 5105 Po wpo 5543 Or wor 5544 × cxp 5631 ‘cfv 6496 1^st c1st 7919 2^nd c2nd 7920 2c2 12208 ♯chash 14230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231

This theorem is referenced by: prproropf1olem3 45687 prproropf1o 45689

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