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Theorem upgr0eopALT 28903
Description: Alternate proof of upgr0eop 28901, using the general theorem gropeld 28820 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 28901). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
upgr0eopALT (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Proof of Theorem upgr0eopALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 vex 3473 . . . . . 6 𝑔 ∈ V
21a1i 11 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V)
3 simpr 484 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅)
42, 3upgr0e 28898 . . . 4 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
54ax-gen 1790 . . 3 𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
65a1i 11 . 2 (𝑉𝑊 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph))
7 id 22 . 2 (𝑉𝑊𝑉𝑊)
8 0ex 5301 . . 3 ∅ ∈ V
98a1i 11 . 2 (𝑉𝑊 → ∅ ∈ V)
106, 7, 9gropeld 28820 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1532   = wceq 1534  wcel 2099  Vcvv 3469  c0 4318  cop 4630  cfv 6542  Vtxcvtx 28783  iEdgciedg 28784  UPGraphcupgr 28867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-i2m1 11192  ax-1ne0 11193  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-1st 7985  df-2nd 7986  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-2 12291  df-vtx 28785  df-iedg 28786  df-upgr 28869  df-umgr 28870
This theorem is referenced by: (None)
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