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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 4305 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐵, 𝐶, 𝐴}) | |
3 | tprot 4284 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
4 | 2, 3 | syl6eleqr 2712 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶}) |
5 | ne0i 3921 | . 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 1990 ≠ wne 2794 ∅c0 3915 {ctp 4181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 df-tp 4182 |
This theorem is referenced by: raltpd 4315 fr3nr 6979 limsupequzlem 39954 etransclem48 40499 |
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