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Mirrors > Home > MPE Home > Th. List > ifpsnprss | Structured version Visualization version GIF version |
Description: Lemma for wlkvtxeledg 27407: 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 3992 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴} ⊆ {𝐴}) | |
2 | preq2 4672 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
3 | dfsn2 4582 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | 2, 3 | syl6eqr 2876 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | eqcoms 2831 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} = {𝐴}) |
7 | simpr 487 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → 𝐸 = {𝐴}) | |
8 | 1, 6, 7 | 3sstr4d 4016 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐸 = {𝐴}) → {𝐴, 𝐵} ⊆ 𝐸) |
9 | 8 | 1fpid3 1075 | 1 ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 if-wif 1057 = wceq 1537 ⊆ wss 3938 {csn 4569 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-sn 4570 df-pr 4572 |
This theorem is referenced by: wlkvtxeledg 27407 |
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