Theorem upgr1wlkdlem2 27006

Description: Lemma 2 for upgr1wlkd 27007. (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 3624	. . 3 ⊢ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}
3		sseq2 3627	. . . 4 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
4	3	adantl 482	. . 3 ⊢ (((𝜑 ∧ 𝑋 ≠ 𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
5	2, 4	mpbiri 248	. 2 ⊢ (((𝜑 ∧ 𝑋 ≠ 𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
6	1, 5	mpidan 704	1 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ⊆ wss 3574  {cpr 4179  ‘cfv 5888  ⟨“cs1 13294  ⟨“cs2 13586  Vtxcvtx 25874  iEdgciedg 25875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by:  upgr1wlkd  27007  upgr1trld  27008  upgr1pthd  27009  upgr1pthond  27010