Theorem upgr1wlkdlem2 27924

Description: Lemma 2 for upgr1wlkd 27925. (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 3988	. . 3 ⊢ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}
3		sseq2 3992	. . . 4 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
4	3	adantl 484	. . 3 ⊢ (((𝜑 ∧ 𝑋 ≠ 𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
5	2, 4	mpbiri 260	. 2 ⊢ (((𝜑 ∧ 𝑋 ≠ 𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
6	1, 5	mpidan 687	1 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   ⊆ wss 3935  {cpr 4568  ‘cfv 6354  ⟨“cs1 13948  ⟨“cs2 14202  Vtxcvtx 26780  iEdgciedg 26781

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 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951

This theorem is referenced by:  upgr1wlkd  27925  upgr1trld  27926  upgr1pthd  27927  upgr1pthond  27928