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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 4633 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | elun 4162 | . . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵})) | |
3 | velsn 4646 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 4646 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
5 | 3, 4 | orbi12i 914 | . . . 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 2874 | . 2 ⊢ ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
8 | 1, 7 | eqtri 2762 | 1 ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 847 = wceq 1536 ∈ wcel 2105 {cab 2711 ∪ cun 3960 {csn 4630 {cpr 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-un 3967 df-sn 4631 df-pr 4633 |
This theorem is referenced by: dfsn2ALT 4651 elprg 4652 nfpr 4696 pwpw0 4817 pwsn 4904 zfpair 5426 grothprimlem 10870 nb3grprlem1 29411 rabsspr 32528 abpr 43398 |
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