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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 4660 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶}) | |
3 | ne0i 4223 | . 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 2114 ≠ wne 2934 ∅c0 4211 {ctp 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-sn 4517 df-pr 4519 df-tp 4521 |
This theorem is referenced by: raltpd 4672 fr3nr 7513 limsupequzlem 42805 etransclem48 43365 |
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