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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 4670 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
2 | dfsn2 4574 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | 1, 2 | eqtr4di 2796 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
4 | 3 | eqcoms 2746 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | eqeq2d 2749 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴})) |
6 | 5 | biimpac 479 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴}) |
7 | eqimss2 3977 | . . 3 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃) | |
8 | 7 | adantr 481 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃) |
9 | 6, 8 | ifpimpda 1080 | 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 1060 = wceq 1539 ⊆ wss 3886 {csn 4561 {cpr 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3431 df-un 3891 df-in 3893 df-ss 3903 df-sn 4562 df-pr 4564 |
This theorem is referenced by: upgriswlk 28017 eupth2lem3lem7 28606 upwlkwlk 45279 |
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