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Theorem usgra2edg1 25912
 Description: If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
Assertion
Ref Expression
usgra2edg1 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
Distinct variable groups:   𝐸,𝑏,𝑐,𝑥   𝑁,𝑏,𝑐,𝑥   𝑥,𝑉
Allowed substitution hints:   𝑉(𝑏,𝑐)

Proof of Theorem usgra2edg1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 usgra2edg 25911 . . . . 5 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)))
2 3simpc 1053 . . . . . . . 8 ((𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → (𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)))
3 df-ne 2782 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
43biimpi 205 . . . . . . . . 9 (𝑥𝑦 → ¬ 𝑥 = 𝑦)
543ad2ant1 1075 . . . . . . . 8 ((𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → ¬ 𝑥 = 𝑦)
62, 5jca 553 . . . . . . 7 ((𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → ((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) ∧ ¬ 𝑥 = 𝑦))
76reximi 2994 . . . . . 6 (∃𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → ∃𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) ∧ ¬ 𝑥 = 𝑦))
87reximi 2994 . . . . 5 (∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) ∧ ¬ 𝑥 = 𝑦))
91, 8syl 17 . . . 4 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) ∧ ¬ 𝑥 = 𝑦))
10 rexanali 2981 . . . . . 6 (∃𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦))
1110rexbii 3023 . . . . 5 (∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑥 ∈ dom 𝐸 ¬ ∀𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦))
12 rexnal 2978 . . . . 5 (∃𝑥 ∈ dom 𝐸 ¬ ∀𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦))
1311, 12bitri 263 . . . 4 (∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦))
149, 13sylib 207 . . 3 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∀𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦))
1514intnand 953 . 2 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ (∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥) ∧ ∀𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦)))
16 fveq2 6103 . . . 4 (𝑥 = 𝑦 → (𝐸𝑥) = (𝐸𝑦))
1716eleq2d 2673 . . 3 (𝑥 = 𝑦 → (𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∈ (𝐸𝑦)))
1817reu4 3367 . 2 (∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥) ↔ (∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥) ∧ ∀𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸((𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)) → 𝑥 = 𝑦)))
1915, 18sylnibr 318 1 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ‘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-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-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:  vdn1frgrav2  26552  vdgn1frgrav2  26553
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