Theorem upgr1wlkdlem2

Description: Lemma 2 for upgr1wlkd . (Contributed by AV, 22-Jan-2021)

	Ref	Expression
Hypotheses	upgr1wlkd.p	⊢ 𝑃 = ⟨“ 𝑋 𝑌 ”⟩
	upgr1wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 ”⟩
	upgr1wlkd.x	⊢ ( 𝜑 → 𝑋 ∈ ( Vtx ‘ 𝐺 ) )
	upgr1wlkd.y	⊢ ( 𝜑 → 𝑌 ∈ ( Vtx ‘ 𝐺 ) )
	upgr1wlkd.j	⊢ ( 𝜑 → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } )
Assertion	upgr1wlkdlem2	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		upgr1wlkd.p	⊢ 𝑃 = ⟨“ 𝑋 𝑌 ”⟩
2		upgr1wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 ”⟩
3		upgr1wlkd.x	⊢ ( 𝜑 → 𝑋 ∈ ( Vtx ‘ 𝐺 ) )
4		upgr1wlkd.y	⊢ ( 𝜑 → 𝑌 ∈ ( Vtx ‘ 𝐺 ) )
5		upgr1wlkd.j	⊢ ( 𝜑 → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } )
6		ssid	⊢ { 𝑋 , 𝑌 } ⊆ { 𝑋 , 𝑌 }
7		sseq2	⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } → ( { 𝑋 , 𝑌 } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) ↔ { 𝑋 , 𝑌 } ⊆ { 𝑋 , 𝑌 } ) )
8	7	adantl	⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } ) → ( { 𝑋 , 𝑌 } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) ↔ { 𝑋 , 𝑌 } ⊆ { 𝑋 , 𝑌 } ) )
9	6 8	mpbiri	⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } ) → { 𝑋 , 𝑌 } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) )
10	5 9	mpidan	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) )