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Mirrors > Home > MPE Home > Th. List > dfpr2 | Structured version Visualization version GIF version |
Description: Alternate definition of unordered 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 4180 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | elun 3753 | . . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵})) | |
3 | velsn 4193 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 4193 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
5 | 3, 4 | orbi12i 543 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
6 | 2, 5 | bitri 264 | . . 3 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
7 | 6 | abbi2i 2738 | . 2 ⊢ ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
8 | 1, 7 | eqtri 2644 | 1 ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 383 = wceq 1483 ∈ wcel 1990 {cab 2608 ∪ cun 3572 {csn 4177 {cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: elprg 4196 nfpr 4232 pwpw0 4344 pwsn 4428 pwsnALT 4429 zfpair 4904 grothprimlem 9655 nb3grprlem1 26282 |
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