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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 35129 | . 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V → (𝐴 × 𝐵) ∈ V)) | |
2 | eldifsni 4728 | . . 3 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → 𝐴 ≠ ∅) | |
3 | bj-xpnzex 35128 | . . 3 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V)) |
5 | 1, 4 | impbid 211 | 1 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2109 ≠ wne 2944 Vcvv 3430 ∖ cdif 3888 ∅c0 4261 {csn 4566 × cxp 5586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-xp 5594 df-rel 5595 df-cnv 5596 df-dm 5598 df-rn 5599 |
This theorem is referenced by: (None) |
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