![]() |
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 4737 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
2 | dfsn2 4640 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | 1, 2 | eqtr4di 2788 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
4 | 3 | eqcoms 2738 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | eqeq2d 2741 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴})) |
6 | 5 | biimpac 477 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴}) |
7 | eqimss2 4040 | . . 3 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃) | |
8 | 7 | adantr 479 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃) |
9 | 6, 8 | ifpimpda 1079 | 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 1059 = wceq 1539 ⊆ wss 3947 {csn 4627 {cpr 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ifp 1060 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-un 3952 df-in 3954 df-ss 3964 df-sn 4628 df-pr 4630 |
This theorem is referenced by: upgriswlk 29165 eupth2lem3lem7 29754 upwlkwlk 46815 |
Copyright terms: Public domain | W3C validator |