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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 5564 | . 2 ⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)} | |
2 | velsn 4586 | . . . 4 ⊢ (𝑎 ∈ {𝑋} ↔ 𝑎 = 𝑋) | |
3 | 2 | anbi1i 625 | . . 3 ⊢ ((𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶) ↔ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)) |
4 | 3 | opabbii 5136 | . 2 ⊢ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ {𝑋} ∧ 𝑏 ∈ 𝐶)} = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} |
5 | 1, 4 | eqtri 2847 | 1 ⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4570 {copab 5131 × cxp 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-sn 4571 df-opab 5132 df-xp 5564 |
This theorem is referenced by: xpiun 44040 |
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