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Theorem 1to2vfriswmgra 26533
 Description: Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
1to2vfriswmgra ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
Distinct variable groups:   𝐴,,𝑣,𝑤   𝐵,,𝑣,𝑤   ,𝐸,𝑣,𝑤   ,𝑉,𝑣,𝑤   𝑣,𝑋,𝑤
Allowed substitution hint:   𝑋()

Proof of Theorem 1to2vfriswmgra
StepHypRef Expression
1 1vwmgra 26530 . . . . 5 ((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))
21a1d 25 . . . 4 ((𝐴𝑋𝑉 = {𝐴}) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
32expcom 450 . . 3 (𝑉 = {𝐴} → (𝐴𝑋 → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
4 breq1 4586 . . . . . . . . 9 (𝑉 = {𝐴, 𝐵} → (𝑉 FriendGrph 𝐸 ↔ {𝐴, 𝐵} FriendGrph 𝐸))
54adantr 480 . . . . . . . 8 ((𝑉 = {𝐴, 𝐵} ∧ ((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋)) → (𝑉 FriendGrph 𝐸 ↔ {𝐴, 𝐵} FriendGrph 𝐸))
6 pm3.22 464 . . . . . . . . . . . 12 (((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋) → (𝐴𝑋 ∧ (𝐵 ∈ V ∧ 𝐴𝐵)))
7 anass 679 . . . . . . . . . . . 12 (((𝐴𝑋𝐵 ∈ V) ∧ 𝐴𝐵) ↔ (𝐴𝑋 ∧ (𝐵 ∈ V ∧ 𝐴𝐵)))
86, 7sylibr 223 . . . . . . . . . . 11 (((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋) → ((𝐴𝑋𝐵 ∈ V) ∧ 𝐴𝐵))
9 frgra2v 26526 . . . . . . . . . . 11 (((𝐴𝑋𝐵 ∈ V) ∧ 𝐴𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
108, 9syl 17 . . . . . . . . . 10 (((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
1110adantl 481 . . . . . . . . 9 ((𝑉 = {𝐴, 𝐵} ∧ ((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋)) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
1211pm2.21d 117 . . . . . . . 8 ((𝑉 = {𝐴, 𝐵} ∧ ((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋)) → ({𝐴, 𝐵} FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
135, 12sylbid 229 . . . . . . 7 ((𝑉 = {𝐴, 𝐵} ∧ ((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋)) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
1413expcom 450 . . . . . 6 (((𝐵 ∈ V ∧ 𝐴𝐵) ∧ 𝐴𝑋) → (𝑉 = {𝐴, 𝐵} → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
1514ex 449 . . . . 5 ((𝐵 ∈ V ∧ 𝐴𝐵) → (𝐴𝑋 → (𝑉 = {𝐴, 𝐵} → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))))
1615com23 84 . . . 4 ((𝐵 ∈ V ∧ 𝐴𝐵) → (𝑉 = {𝐴, 𝐵} → (𝐴𝑋 → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))))
17 ianor 508 . . . . . . 7 (¬ (𝐵 ∈ V ∧ 𝐴𝐵) ↔ (¬ 𝐵 ∈ V ∨ ¬ 𝐴𝐵))
18 prprc2 4244 . . . . . . . 8 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
19 nne 2786 . . . . . . . . 9 𝐴𝐵𝐴 = 𝐵)
20 preq2 4213 . . . . . . . . . . 11 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2120eqcoms 2618 . . . . . . . . . 10 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴, 𝐴})
22 dfsn2 4138 . . . . . . . . . 10 {𝐴} = {𝐴, 𝐴}
2321, 22syl6eqr 2662 . . . . . . . . 9 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2419, 23sylbi 206 . . . . . . . 8 𝐴𝐵 → {𝐴, 𝐵} = {𝐴})
2518, 24jaoi 393 . . . . . . 7 ((¬ 𝐵 ∈ V ∨ ¬ 𝐴𝐵) → {𝐴, 𝐵} = {𝐴})
2617, 25sylbi 206 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐴𝐵) → {𝐴, 𝐵} = {𝐴})
2726eqeq2d 2620 . . . . 5 (¬ (𝐵 ∈ V ∧ 𝐴𝐵) → (𝑉 = {𝐴, 𝐵} ↔ 𝑉 = {𝐴}))
2827, 3syl6bi 242 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴𝐵) → (𝑉 = {𝐴, 𝐵} → (𝐴𝑋 → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))))
2916, 28pm2.61i 175 . . 3 (𝑉 = {𝐴, 𝐵} → (𝐴𝑋 → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
303, 29jaoi 393 . 2 ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) → (𝐴𝑋 → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
3130impcom 445 1 ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   FriendGrph cfrgra 26515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-frgra 26516 This theorem is referenced by:  1to3vfriswmgra  26534
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