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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 4734 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
| 2 | dfsn2 4639 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
| 3 | 1, 2 | eqtr4di 2795 | . . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴}) |
| 4 | 3 | eqcoms 2745 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
| 5 | 4 | eqeq2d 2748 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑃 = {𝐴, 𝐵} ↔ 𝑃 = {𝐴})) |
| 6 | 5 | biimpac 478 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ 𝐴 = 𝐵) → 𝑃 = {𝐴}) |
| 7 | eqimss2 4043 | . . 3 ⊢ (𝑃 = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ 𝑃) | |
| 8 | 7 | adantr 480 | . 2 ⊢ ((𝑃 = {𝐴, 𝐵} ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ⊆ 𝑃) |
| 9 | 6, 8 | ifpimpda 1081 | 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 1063 = wceq 1540 ⊆ wss 3951 {csn 4626 {cpr 4628 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: upgriswlk 29659 eupth2lem3lem7 30253 upwlkwlk 48055 |
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