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Mirrors > Home > MPE Home > Th. List > tpnzd | Structured version Visualization version GIF version |
Description: A triplet 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 | tpid3g 4248 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐵, 𝐶, 𝐴}) | |
3 | tprot 4228 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
4 | 2, 3 | syl6eleqr 2699 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶}) |
5 | ne0i 3880 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅) | |
6 | 1, 4, 5 | 3syl 18 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 {ctp 4129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 df-tp 4130 |
This theorem is referenced by: raltpd 4258 fr3nr 6871 etransclem48 39175 |
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