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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpexb | Structured version Visualization version GIF version |
Description: A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.) |
Ref | Expression |
---|---|
xpexb | ⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvxp 6178 | . . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
2 | cnvexg 7946 | . . 3 ⊢ ((𝐴 × 𝐵) ∈ V → ◡(𝐴 × 𝐵) ∈ V) | |
3 | 1, 2 | eqeltrrid 2843 | . 2 ⊢ ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V) |
4 | cnvxp 6178 | . . 3 ⊢ ◡(𝐵 × 𝐴) = (𝐴 × 𝐵) | |
5 | cnvexg 7946 | . . 3 ⊢ ((𝐵 × 𝐴) ∈ V → ◡(𝐵 × 𝐴) ∈ V) | |
6 | 4, 5 | eqeltrrid 2843 | . 2 ⊢ ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V) |
7 | 3, 6 | impbii 209 | 1 ⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2105 Vcvv 3477 × cxp 5686 ◡ccnv 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 |
This theorem is referenced by: (None) |
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