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Mirrors > Home > MPE Home > Th. List > upgr1wlkdlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for upgr1wlkd 29400. (Contributed by AV, 22-Jan-2021.) |
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}) |
Ref | Expression |
---|---|
upgr1wlkdlem2 | ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgr1wlkd.j | . 2 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) | |
2 | ssid 4005 | . . 3 ⊢ {𝑋, 𝑌} ⊆ {𝑋, 𝑌} | |
3 | sseq2 4009 | . . . 4 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌})) | |
4 | 3 | adantl 483 | . . 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 688 | 1 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ⊆ wss 3949 {cpr 4631 ‘cfv 6544 ⟨“cs1 14545 ⟨“cs2 14792 Vtxcvtx 28256 iEdgciedg 28257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 |
This theorem is referenced by: upgr1wlkd 29400 upgr1trld 29401 upgr1pthd 29402 upgr1pthond 29403 |
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