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Mirrors > Home > MPE Home > Th. List > ifpsnprss | Structured version Visualization version GIF version |
Description: Lemma for wlkvtxeledg 28919: 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 4005 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴}) | |
2 | preq2 4738 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
3 | dfsn2 4641 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | 2, 3 | eqtr4di 2790 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | eqcoms 2740 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴}) |
7 | simpr 485 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → 𝐸 = {𝐴}) | |
8 | 1, 6, 7 | 3sstr4d 4029 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸) |
9 | 8 | 1fpid3 1082 | 1 ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 if-wif 1061 = wceq 1541 ⊆ wss 3948 {csn 4628 {cpr 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-pr 4631 |
This theorem is referenced by: wlkvtxeledg 28919 |
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