Theorem usgrnloopALT

Description: Alternate proof of usgrnloop , not using umgrnloop . (Contributed by Alexander van der Vekens, 19-Aug-2017) (Proof shortened by Alexander van der Vekens, 20-Mar-2018) (Revised by AV, 17-Oct-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	usgrnloopv.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	usgrnloopALT	⊢ ( 𝐺 ∈ USGraph → ( ∃ 𝑥 ∈ dom 𝐸 ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		usgrnloopv.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3	1 2	usgredgprv	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) ) )
4	3	imp	⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) )
5	1	usgrnloopv	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )
6	5	ex	⊢ ( 𝐺 ∈ USGraph → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) ) )
7	6	com23	⊢ ( 𝐺 ∈ USGraph → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → 𝑀 ≠ 𝑁 ) ) )
8	7	adantr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → 𝑀 ≠ 𝑁 ) ) )
9	8	imp	⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → 𝑀 ≠ 𝑁 ) )
10	9	com12	⊢ ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → ( ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → 𝑀 ≠ 𝑁 ) )
11	10	adantr	⊢ ( ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → 𝑀 ≠ 𝑁 ) )
12	4 11	mpcom	⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → 𝑀 ≠ 𝑁 )
13	12	ex	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑥 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )
14	13	rexlimdva	⊢ ( 𝐺 ∈ USGraph → ( ∃ 𝑥 ∈ dom 𝐸 ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )