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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelopabid | Structured version Visualization version GIF version |
Description: Membership in an ordered-pair class abstraction. One can remove the DV condition on 𝑥, 𝑦 by using opabid 5376 in place of opabidw 5375. (Contributed by BJ, 22-May-2024.) |
Ref | Expression |
---|---|
bj-opelopabid | ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5026 | . 2 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | opabidw 5375 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
3 | 1, 2 | bitri 278 | 1 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2112 〈cop 4521 class class class wbr 5025 {copab 5087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-v 3409 df-dif 3857 df-un 3859 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-br 5026 df-opab 5088 |
This theorem is referenced by: bj-opabco 34868 |
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