![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4526 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐵, 𝐶, 𝐴}) | |
3 | tprot 4503 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
4 | 2, 3 | syl6eleqr 2918 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶}) |
5 | ne0i 4151 | . 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 2166 ≠ wne 3000 ∅c0 4145 {ctp 4402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-v 3417 df-dif 3802 df-un 3804 df-nul 4146 df-sn 4399 df-pr 4401 df-tp 4403 |
This theorem is referenced by: raltpd 4534 fr3nr 7241 limsupequzlem 40750 etransclem48 41294 |
Copyright terms: Public domain | W3C validator |