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Mirrors > Home > ILE Home > Th. List > tpnz | GIF version |
Description: A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.) |
Ref | Expression |
---|---|
tpnz.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
tpnz | ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpnz.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | tpid1 3715 | . 2 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶} |
3 | ne0i 3441 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2158 ≠ wne 2357 Vcvv 2749 ∅c0 3434 {ctp 3606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-v 2751 df-dif 3143 df-un 3145 df-nul 3435 df-sn 3610 df-pr 3611 df-tp 3612 |
This theorem is referenced by: (None) |
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