Theorem upgr0eopALT 26895

Description: Alternate proof of upgr0eop 26893, using the general theorem gropeld 26812 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 26893). (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 3498	. . . . . 6 ⊢ 𝑔 ∈ V
2	1	a1i 11	. . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V)
3		simpr 487	. . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅)
4	2, 3	upgr0e 26890	. . . 4 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
5	4	ax-gen 1792	. . 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 5204	. . 3 ⊢ ∅ ∈ V
9	8	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → ∅ ∈ V)
10	6, 7, 9	gropeld 26812	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  Vcvv 3495  ∅c0 4291  ⟨cop 4567  ‘cfv 6350  Vtxcvtx 26775  iEdgciedg 26776  UPGraphcupgr 26859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-i2m1 10599  ax-1ne0 10600  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-1st 7683  df-2nd 7684  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-2 11694  df-vtx 26777  df-iedg 26778  df-upgr 26861  df-umgr 26862

This theorem is referenced by: (None)