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Theorem prproropf1olem1 44437
 Description: Lemma 1 for prproropf1o 44441. (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 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)
Distinct variable groups:   𝑉,𝑝   𝑊,𝑝
Allowed substitution hints:   𝑃(𝑝)   𝑅(𝑝)   𝑂(𝑝)

Proof of Theorem prproropf1olem1
StepHypRef Expression
1 prproropf1o.o . . . 4 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
21prproropf1olem0 44436 . . 3 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
3 simpr2 1192 . . . . 5 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))
4 prelpwi 5312 . . . . 5 (((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) → {(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉)
53, 4syl 17 . . . 4 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → {(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉)
6 sopo 5465 . . . . . . 7 (𝑅 Or 𝑉𝑅 Po 𝑉)
76adantr 484 . . . . . 6 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → 𝑅 Po 𝑉)
8 simpr3 1193 . . . . . 6 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (1st𝑊)𝑅(2nd𝑊))
9 po2ne 5462 . . . . . 6 ((𝑅 Po 𝑉 ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)) → (1st𝑊) ≠ (2nd𝑊))
107, 3, 8, 9syl3anc 1368 . . . . 5 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (1st𝑊) ≠ (2nd𝑊))
11 fvex 6676 . . . . . 6 (1st𝑊) ∈ V
12 fvex 6676 . . . . . 6 (2nd𝑊) ∈ V
13 hashprg 13819 . . . . . 6 (((1st𝑊) ∈ V ∧ (2nd𝑊) ∈ V) → ((1st𝑊) ≠ (2nd𝑊) ↔ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
1411, 12, 13mp2an 691 . . . . 5 ((1st𝑊) ≠ (2nd𝑊) ↔ (♯‘{(1st𝑊), (2nd𝑊)}) = 2)
1510, 14sylib 221 . . . 4 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (♯‘{(1st𝑊), (2nd𝑊)}) = 2)
165, 15jca 515 . . 3 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ({(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
172, 16sylan2b 596 . 2 ((𝑅 Or 𝑉𝑊𝑂) → ({(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
18 fveqeq2 6672 . . 3 (𝑝 = {(1st𝑊), (2nd𝑊)} → ((♯‘𝑝) = 2 ↔ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
19 prproropf1o.p . . 3 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
2018, 19elrab2 3607 . 2 ({(1st𝑊), (2nd𝑊)} ∈ 𝑃 ↔ ({(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
2117, 20sylibr 237 1 ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  {crab 3074  Vcvv 3409   ∩ cin 3859  𝒫 cpw 4497  {cpr 4527  ⟨cop 4531   class class class wbr 5036   Po wpo 5445   Or wor 5446   × cxp 5526  ‘cfv 6340  1st c1st 7697  2nd c2nd 7698  2c2 11742  ♯chash 13753 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-oadd 8122  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-dju 9376  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-n0 11948  df-z 12034  df-uz 12296  df-fz 12953  df-hash 13754 This theorem is referenced by:  prproropf1olem3  44439  prproropf1o  44441
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