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Mirrors > Home > MPE Home > Th. List > Mathboxes > opelxpii | Structured version Visualization version GIF version |
Description: Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.) |
Ref | Expression |
---|---|
opelxpii.1 | ⊢ 𝐴 ∈ 𝐶 |
opelxpii.2 | ⊢ 𝐵 ∈ 𝐷 |
Ref | Expression |
---|---|
opelxpii | ⊢ 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpii.1 | . 2 ⊢ 𝐴 ∈ 𝐶 | |
2 | opelxpii.2 | . 2 ⊢ 𝐵 ∈ 𝐷 | |
3 | opelxpi 5585 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 〈cop 4566 × cxp 5546 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-xp 5554 |
This theorem is referenced by: (None) |
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