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Mirrors > Home > MPE Home > Th. List > opelxpii | Structured version Visualization version GIF version |
Description: Ordered pair membership in a Cartesian product (implication), induction form. (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 5719 | . 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 2098 ⟨cop 4638 × cxp 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-opab 5215 df-xp 5688 |
This theorem is referenced by: pzriprnglem7 21420 pzriprng1ALT 21429 |
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