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Mirrors > Home > MPE Home > Th. List > ifpsnprss | Structured version Visualization version GIF version |
Description: Lemma for wlkvtxeledg 29657: Two adjacent (not necessarily different) vertices 𝐴 and 𝐵 in a walk are incident with an edge 𝐸. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.) |
Ref | Expression |
---|---|
ifpsnprss | ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 4019 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴}) | |
2 | preq2 4739 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
3 | dfsn2 4644 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | 2, 3 | eqtr4di 2793 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | eqcoms 2743 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴}) |
7 | simpr 484 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → 𝐸 = {𝐴}) | |
8 | 1, 6, 7 | 3sstr4d 4043 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸) |
9 | 8 | 1fpid3 1081 | 1 ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 if-wif 1062 = wceq 1537 ⊆ wss 3963 {csn 4631 {cpr 4633 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 |
This theorem is referenced by: wlkvtxeledg 29657 |
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