| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > dfpr2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of a pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
| Ref | Expression |
|---|---|
| dfpr2 | ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pr 4629 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 2 | elun 4153 | . . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵})) | |
| 3 | velsn 4642 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
| 4 | velsn 4642 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
| 5 | 3, 4 | orbi12i 915 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
| 6 | 2, 5 | bitri 275 | . . 3 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
| 7 | 6 | eqabi 2877 | . 2 ⊢ ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
| 8 | 1, 7 | eqtri 2765 | 1 ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
| Colors of variables: wff setvar class |
| Syntax hints: ∨ wo 848 = wceq 1540 ∈ wcel 2108 {cab 2714 ∪ cun 3949 {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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: dfsn2ALT 4647 elprg 4648 nfpr 4692 pwpw0 4813 pwsn 4900 zfpair 5421 grothprimlem 10873 nb3grprlem1 29397 rabsspr 32520 abpr 43422 |
| Copyright terms: Public domain | W3C validator |