![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ifpprsnss | Structured version Visualization version GIF version |
Description: An unordered pair is a singleton or a subset of itself. This theorem is helpful to convert theorems about walks in arbitrary graphs into theorems about walks in pseudographs. (Contributed by AV, 27-Feb-2021.) |
Ref | Expression |
---|---|
ifpprsnss | ⊢ (𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 4500 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
2 | dfsn2 4410 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | 1, 2 | syl6eqr 2831 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
4 | 3 | eqcoms 2785 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | eqeq2d 2787 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴})) |
6 | 5 | biimpac 472 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴}) |
7 | eqimss2 3876 | . . 3 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃) | |
8 | 7 | adantr 474 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃) |
9 | 6, 8 | ifpimpda 1062 | 1 ⊢ (𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 if-wif 1046 = wceq 1601 ⊆ wss 3791 {csn 4397 {cpr 4399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ifp 1047 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-v 3399 df-un 3796 df-in 3798 df-ss 3805 df-sn 4398 df-pr 4400 |
This theorem is referenced by: upgriswlk 26988 eupth2lem3lem7 27638 upwlkwlk 42755 |
Copyright terms: Public domain | W3C validator |