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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 4636 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | elun 4148 | . . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵})) | |
3 | velsn 4649 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 4649 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
5 | 3, 4 | orbi12i 912 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
6 | 2, 5 | bitri 274 | . . 3 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
7 | 6 | eqabi 2862 | . 2 ⊢ ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
8 | 1, 7 | eqtri 2754 | 1 ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1534 ∈ wcel 2099 {cab 2703 ∪ cun 3945 {csn 4633 {cpr 4635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-un 3952 df-sn 4634 df-pr 4636 |
This theorem is referenced by: dfsn2ALT 4654 elprg 4655 nfpr 4699 pwpw0 4822 pwsn 4906 zfpair 5425 grothprimlem 10876 nb3grprlem1 29316 abpr 43076 |
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