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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpsnopab | Structured version Visualization version GIF version |
Description: A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
Ref | Expression |
---|---|
xpsnopab | ⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 5044 | . 2 ⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)} | |
2 | velsn 4141 | . . . 4 ⊢ (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋) | |
3 | 2 | anbi1i 727 | . . 3 ⊢ ((𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶) ↔ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)) |
4 | 3 | opabbii 4649 | . 2 ⊢ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)} = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} |
5 | 1, 4 | eqtri 2632 | 1 ⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 {copab 4642 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sn 4126 df-opab 4644 df-xp 5044 |
This theorem is referenced by: xpiun 41556 |
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