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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 3718 | . 2 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶} |
3 | ne0i 3444 | . 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 2160 ≠ wne 2360 Vcvv 2752 ∅c0 3437 {ctp 3609 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-v 2754 df-dif 3146 df-un 3148 df-nul 3438 df-sn 3613 df-pr 3614 df-tp 3615 |
This theorem is referenced by: (None) |
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