Theorem upgr0eopALT 27467

Description: Alternate proof of upgr0eop 27465, using the general theorem gropeld 27384 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 27465). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
upgr0eopALT	⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Proof of Theorem upgr0eopALT

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3434	. . . . . 6 ⊢ 𝑔 ∈ V
2	1	a1i 11	. . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V)
3		simpr 484	. . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅)
4	2, 3	upgr0e 27462	. . . 4 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
5	4	ax-gen 1801	. . 3 ⊢ ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
6	5	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph))
7		id 22	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ 𝑊)
8		0ex 5234	. . 3 ⊢ ∅ ∈ V
9	8	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → ∅ ∈ V)
10	6, 7, 9	gropeld 27384	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ∅c0 4261  ⟨cop 4572  ‘cfv 6430  Vtxcvtx 27347  iEdgciedg 27348  UPGraphcupgr 27431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-i2m1 10923  ax-1ne0 10924  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-1st 7817  df-2nd 7818  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-2 12019  df-vtx 27349  df-iedg 27350  df-upgr 27433  df-umgr 27434

This theorem is referenced by: (None)