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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 4739 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
2 | dfsn2 4644 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | 1, 2 | eqtr4di 2793 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
4 | 3 | eqcoms 2743 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | eqeq2d 2746 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴})) |
6 | 5 | biimpac 478 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴}) |
7 | eqimss2 4055 | . . 3 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃) | |
8 | 7 | adantr 480 | . 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 1062 = wceq 1537 ⊆ wss 3963 {csn 4631 {cpr 4633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 |
This theorem is referenced by: upgriswlk 29674 eupth2lem3lem7 30263 upwlkwlk 47983 |
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