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Theorem upgr0eopALT 28971
Description: Alternate proof of upgr0eop 28969, using the general theorem gropeld 28888 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 28969). (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 3467 . . . . . 6 𝑔 ∈ V
21a1i 11 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V)
3 simpr 483 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅)
42, 3upgr0e 28966 . . . 4 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
54ax-gen 1789 . . 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 5302 . . 3 ∅ ∈ V
98a1i 11 . 2 (𝑉𝑊 → ∅ ∈ V)
106, 7, 9gropeld 28888 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531   = wceq 1533  wcel 2098  Vcvv 3463  c0 4318  cop 4630  cfv 6542  Vtxcvtx 28851  iEdgciedg 28852  UPGraphcupgr 28935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-i2m1 11204  ax-1ne0 11205  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  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 7418  df-1st 7989  df-2nd 7990  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-2 12303  df-vtx 28853  df-iedg 28854  df-upgr 28937  df-umgr 28938
This theorem is referenced by: (None)
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