| 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 6177 | . . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
| 2 | cnvexg 7946 | . . 3 ⊢ ((𝐴 × 𝐵) ∈ V → ◡(𝐴 × 𝐵) ∈ V) | |
| 3 | 1, 2 | eqeltrrid 2846 | . 2 ⊢ ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V) |
| 4 | cnvxp 6177 | . . 3 ⊢ ◡(𝐵 × 𝐴) = (𝐴 × 𝐵) | |
| 5 | cnvexg 7946 | . . 3 ⊢ ((𝐵 × 𝐴) ∈ V → ◡(𝐵 × 𝐴) ∈ V) | |
| 6 | 4, 5 | eqeltrrid 2846 | . 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 2108 Vcvv 3480 × cxp 5683 ◡ccnv 5684 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |