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Mirrors > Home > MPE Home > Th. List > tpnzd | Structured version Visualization version GIF version |
Description: An unordered triple containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.) |
Ref | Expression |
---|---|
tpnzd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
tpnzd | ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpnzd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | tpid1g 4705 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶}) | |
3 | ne0i 4268 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅) | |
4 | 1, 2, 3 | 3syl 18 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 {ctp 4565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-sn 4562 df-pr 4564 df-tp 4566 |
This theorem is referenced by: raltpd 4717 fr3nr 7622 limsupequzlem 43263 etransclem48 43823 |
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