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Theorem prproropf1olem1 45769
Description: Lemma 1 for prproropf1o 45773. (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 45768 . . 3 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
3 simpr2 1196 . . . . 5 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))
4 prelpwi 5409 . . . . 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 5569 . . . . . . 7 (𝑅 Or 𝑉𝑅 Po 𝑉)
76adantr 482 . . . . . 6 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → 𝑅 Po 𝑉)
8 simpr3 1197 . . . . . 6 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (1st𝑊)𝑅(2nd𝑊))
9 po2ne 5566 . . . . . 6 ((𝑅 Po 𝑉 ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)) → (1st𝑊) ≠ (2nd𝑊))
107, 3, 8, 9syl3anc 1372 . . . . 5 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (1st𝑊) ≠ (2nd𝑊))
11 fvex 6860 . . . . . 6 (1st𝑊) ∈ V
12 fvex 6860 . . . . . 6 (2nd𝑊) ∈ V
13 hashprg 14302 . . . . . 6 (((1st𝑊) ∈ V ∧ (2nd𝑊) ∈ V) → ((1st𝑊) ≠ (2nd𝑊) ↔ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
1411, 12, 13mp2an 691 . . . . 5 ((1st𝑊) ≠ (2nd𝑊) ↔ (♯‘{(1st𝑊), (2nd𝑊)}) = 2)
1510, 14sylib 217 . . . 4 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → (♯‘{(1st𝑊), (2nd𝑊)}) = 2)
165, 15jca 513 . . 3 ((𝑅 Or 𝑉 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊))) → ({(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
172, 16sylan2b 595 . 2 ((𝑅 Or 𝑉𝑊𝑂) → ({(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
18 fveqeq2 6856 . . 3 (𝑝 = {(1st𝑊), (2nd𝑊)} → ((♯‘𝑝) = 2 ↔ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
19 prproropf1o.p . . 3 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
2018, 19elrab2 3653 . 2 ({(1st𝑊), (2nd𝑊)} ∈ 𝑃 ↔ ({(1st𝑊), (2nd𝑊)} ∈ 𝒫 𝑉 ∧ (♯‘{(1st𝑊), (2nd𝑊)}) = 2))
2117, 20sylibr 233 1 ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2944  {crab 3410  Vcvv 3448  cin 3914  𝒫 cpw 4565  {cpr 4593  cop 4597   class class class wbr 5110   Po wpo 5548   Or wor 5549   × cxp 5636  cfv 6501  1st c1st 7924  2nd c2nd 7925  2c2 12215  chash 14237
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-hash 14238
This theorem is referenced by:  prproropf1olem3  45771  prproropf1o  45773
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