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Theorem elusuhgra 25897
 Description: An undirected simple graph without loops is an undirected hypergraph. (Contributed by AV, 9-Jan-2020.)
Assertion
Ref Expression
elusuhgra (𝐺 ∈ USGrph → 𝐺 ∈ UHGrph )

Proof of Theorem elusuhgra
StepHypRef Expression
1 relusgra 25864 . . 3 Rel USGrph
2 1st2nd 7105 . . 3 ((Rel USGrph ∧ 𝐺 ∈ USGrph ) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
31, 2mpan 702 . 2 (𝐺 ∈ USGrph → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
4 usisuhgra 25896 . . . 4 ((1st𝐺) USGrph (2nd𝐺) → (1st𝐺) UHGrph (2nd𝐺))
54a1i 11 . . 3 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → ((1st𝐺) USGrph (2nd𝐺) → (1st𝐺) UHGrph (2nd𝐺)))
6 eleq1 2676 . . . 4 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ USGrph ))
7 df-br 4584 . . . 4 ((1st𝐺) USGrph (2nd𝐺) ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ USGrph )
86, 7syl6bbr 277 . . 3 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph ↔ (1st𝐺) USGrph (2nd𝐺)))
9 eleq1 2676 . . . 4 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ UHGrph ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ UHGrph ))
10 df-br 4584 . . . 4 ((1st𝐺) UHGrph (2nd𝐺) ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ UHGrph )
119, 10syl6bbr 277 . . 3 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ UHGrph ↔ (1st𝐺) UHGrph (2nd𝐺)))
125, 8, 113imtr4d 282 . 2 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph → 𝐺 ∈ UHGrph ))
133, 12mpcom 37 1 (𝐺 ∈ USGrph → 𝐺 ∈ UHGrph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  Rel wrel 5043  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058   UHGrph cuhg 25819   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-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890 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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-2 10956  df-uhgra 25821  df-umgra 25842  df-uslgra 25861  df-usgra 25862 This theorem is referenced by:  0eusgraiff0rgracl  26468
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