Theorem upgr1wlkdlem2 29868

Description: Lemma 2 for upgr1wlkd 29869. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
upgr1wlkd.p	⊢ 𝑃 = ⟨“𝑋𝑌”⟩
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

Ref	Expression
upgr1wlkdlem2	⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))

Proof of Theorem upgr1wlkdlem2

Step	Hyp	Ref	Expression
1		upgr1wlkd.j	. 2 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
2		ssid 3996	. . 3 ⊢ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}
3		sseq2 4000	. . . 4 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
4	3	adantl 481	. . 3 ⊢ (((𝜑 ∧ 𝑋 ≠ 𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
5	2, 4	mpbiri 258	. 2 ⊢ (((𝜑 ∧ 𝑋 ≠ 𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
6	1, 5	mpidan 686	1 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2932   ⊆ wss 3940  {cpr 4622  ‘cfv 6533  ⟨“cs1 14542  ⟨“cs2 14789  Vtxcvtx 28725  iEdgciedg 28726

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-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3947  df-ss 3957

This theorem is referenced by:  upgr1wlkd  29869  upgr1trld  29870  upgr1pthd  29871  upgr1pthond  29872