![]() |
Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
|
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 6163 | . . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
2 | cnvexg 7932 | . . 3 ⊢ ((𝐴 × 𝐵) ∈ V → ◡(𝐴 × 𝐵) ∈ V) | |
3 | 1, 2 | eqeltrrid 2830 | . 2 ⊢ ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V) |
4 | cnvxp 6163 | . . 3 ⊢ ◡(𝐵 × 𝐴) = (𝐴 × 𝐵) | |
5 | cnvexg 7932 | . . 3 ⊢ ((𝐵 × 𝐴) ∈ V → ◡(𝐵 × 𝐴) ∈ V) | |
6 | 4, 5 | eqeltrrid 2830 | . 2 ⊢ ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V) |
7 | 3, 6 | impbii 208 | 1 ⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 Vcvv 3461 × cxp 5676 ◡ccnv 5677 |
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-11 2146 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |