![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > upgr1wlkdlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for upgr1wlkd 29977. (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 4004 | . . 3 ⊢ {𝑋, 𝑌} ⊆ {𝑋, 𝑌} | |
3 | sseq2 4008 | . . . 4 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌})) | |
4 | 3 | adantl 480 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ≠ 𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌})) |
5 | 2, 4 | mpbiri 257 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ⊆ wss 3949 {cpr 4634 ‘cfv 6553 〈“cs1 14585 〈“cs2 14832 Vtxcvtx 28829 iEdgciedg 28830 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-in 3956 df-ss 3966 |
This theorem is referenced by: upgr1wlkd 29977 upgr1trld 29978 upgr1pthd 29979 upgr1pthond 29980 |
Copyright terms: Public domain | W3C validator |