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Theorem usgra2edg 25911
 Description: If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
Assertion
Ref Expression
usgra2edg (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)))
Distinct variable groups:   𝐸,𝑏,𝑐,𝑥,𝑦   𝑁,𝑏,𝑐,𝑥,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝑉(𝑏,𝑐)

Proof of Theorem usgra2edg
StepHypRef Expression
1 usgrafun 25878 . . . . . 6 (𝑉 USGrph 𝐸 → Fun 𝐸)
2 funfn 5833 . . . . . 6 (Fun 𝐸𝐸 Fn dom 𝐸)
31, 2sylib 207 . . . . 5 (𝑉 USGrph 𝐸𝐸 Fn dom 𝐸)
433ad2ant1 1075 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → 𝐸 Fn dom 𝐸)
5 fvelrnb 6153 . . . . 5 (𝐸 Fn dom 𝐸 → ({𝑁, 𝑏} ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏}))
6 fvelrnb 6153 . . . . 5 (𝐸 Fn dom 𝐸 → ({𝑐, 𝑁} ∈ ran 𝐸 ↔ ∃𝑦 ∈ dom 𝐸(𝐸𝑦) = {𝑐, 𝑁}))
75, 6anbi12d 743 . . . 4 (𝐸 Fn dom 𝐸 → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) ↔ (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} ∧ ∃𝑦 ∈ dom 𝐸(𝐸𝑦) = {𝑐, 𝑁})))
84, 7syl 17 . . 3 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) ↔ (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} ∧ ∃𝑦 ∈ dom 𝐸(𝐸𝑦) = {𝑐, 𝑁})))
9 reeanv 3086 . . . 4 (∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) ↔ (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} ∧ ∃𝑦 ∈ dom 𝐸(𝐸𝑦) = {𝑐, 𝑁}))
10 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐸𝑥) = (𝐸𝑦))
1110eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐸𝑥) = {𝑁, 𝑏} ↔ (𝐸𝑦) = {𝑁, 𝑏}))
1211anbi1d 737 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) ↔ ((𝐸𝑦) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})))
13 eqtr2 2630 . . . . . . . . . . . . . 14 (((𝐸𝑦) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → {𝑁, 𝑏} = {𝑐, 𝑁})
14 prcom 4211 . . . . . . . . . . . . . . . 16 {𝑐, 𝑁} = {𝑁, 𝑐}
1514eqeq2i 2622 . . . . . . . . . . . . . . 15 ({𝑁, 𝑏} = {𝑐, 𝑁} ↔ {𝑁, 𝑏} = {𝑁, 𝑐})
16 vex 3176 . . . . . . . . . . . . . . . . 17 𝑏 ∈ V
17 vex 3176 . . . . . . . . . . . . . . . . 17 𝑐 ∈ V
1816, 17preqr2 4321 . . . . . . . . . . . . . . . 16 ({𝑁, 𝑏} = {𝑁, 𝑐} → 𝑏 = 𝑐)
19 eqneqall 2793 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → (𝑏𝑐𝑥𝑦))
2018, 19syl 17 . . . . . . . . . . . . . . 15 ({𝑁, 𝑏} = {𝑁, 𝑐} → (𝑏𝑐𝑥𝑦))
2115, 20sylbi 206 . . . . . . . . . . . . . 14 ({𝑁, 𝑏} = {𝑐, 𝑁} → (𝑏𝑐𝑥𝑦))
2213, 21syl 17 . . . . . . . . . . . . 13 (((𝐸𝑦) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → (𝑏𝑐𝑥𝑦))
2312, 22syl6bi 242 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → (𝑏𝑐𝑥𝑦)))
2423com23 84 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑏𝑐 → (((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → 𝑥𝑦)))
2524impd 446 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑏𝑐 ∧ ((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})) → 𝑥𝑦))
26 ax-1 6 . . . . . . . . . 10 (𝑥𝑦 → ((𝑏𝑐 ∧ ((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})) → 𝑥𝑦))
2725, 26pm2.61ine 2865 . . . . . . . . 9 ((𝑏𝑐 ∧ ((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})) → 𝑥𝑦)
28273ad2antl3 1218 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})) → 𝑥𝑦)
29 prid1g 4239 . . . . . . . . . . . 12 (𝑁𝑉𝑁 ∈ {𝑁, 𝑏})
30293ad2ant2 1076 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → 𝑁 ∈ {𝑁, 𝑏})
31 eleq2 2677 . . . . . . . . . . 11 ((𝐸𝑥) = {𝑁, 𝑏} → (𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∈ {𝑁, 𝑏}))
3230, 31syl5ibr 235 . . . . . . . . . 10 ((𝐸𝑥) = {𝑁, 𝑏} → ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → 𝑁 ∈ (𝐸𝑥)))
3332adantr 480 . . . . . . . . 9 (((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → 𝑁 ∈ (𝐸𝑥)))
3433impcom 445 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})) → 𝑁 ∈ (𝐸𝑥))
35 prid2g 4240 . . . . . . . . . . . 12 (𝑁𝑉𝑁 ∈ {𝑐, 𝑁})
36353ad2ant2 1076 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → 𝑁 ∈ {𝑐, 𝑁})
37 eleq2 2677 . . . . . . . . . . 11 ((𝐸𝑦) = {𝑐, 𝑁} → (𝑁 ∈ (𝐸𝑦) ↔ 𝑁 ∈ {𝑐, 𝑁}))
3836, 37syl5ibr 235 . . . . . . . . . 10 ((𝐸𝑦) = {𝑐, 𝑁} → ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → 𝑁 ∈ (𝐸𝑦)))
3938adantl 481 . . . . . . . . 9 (((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → 𝑁 ∈ (𝐸𝑦)))
4039impcom 445 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})) → 𝑁 ∈ (𝐸𝑦))
4128, 34, 403jca 1235 . . . . . . 7 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁})) → (𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)))
4241ex 449 . . . . . 6 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → (((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → (𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦))))
4342reximdv 2999 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → (∃𝑦 ∈ dom 𝐸((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → ∃𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦))))
4443reximdv 2999 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → (∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝐸𝑥) = {𝑁, 𝑏} ∧ (𝐸𝑦) = {𝑐, 𝑁}) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦))))
459, 44syl5bir 232 . . 3 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → ((∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} ∧ ∃𝑦 ∈ dom 𝐸(𝐸𝑦) = {𝑐, 𝑁}) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦))))
468, 45sylbid 229 . 2 ((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦))))
4746imp 444 1 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804   USGrph cusg 25859 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-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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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 This theorem is referenced by:  usgra2edg1  25912
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