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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-xpnzexb | Structured version Visualization version GIF version | ||
| Description: If the first factor of a product is a nonempty set, then the product is a set if and only if the second factor is a set. (Contributed by BJ, 2-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| bj-xpnzexb | ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-xpexg2 36962 | . 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V → (𝐴 × 𝐵) ∈ V)) | |
| 2 | eldifsni 4789 | . . 3 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → 𝐴 ≠ ∅) | |
| 3 | bj-xpnzex 36961 | . . 3 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V)) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V)) | 
| 5 | 1, 4 | impbid 212 | 1 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∖ cdif 3947 ∅c0 4332 {csn 4625 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: (None) | 
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