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Mirrors > Home > ILE Home > Th. List > tr0 | GIF version |
Description: The empty set is transitive. (Contributed by NM, 16-Sep-1993.) |
Ref | Expression |
---|---|
tr0 | ⊢ Tr ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3476 | . 2 ⊢ ∅ ⊆ 𝒫 ∅ | |
2 | dftr4 4124 | . 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅) | |
3 | 1, 2 | mpbir 146 | 1 ⊢ Tr ∅ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3144 ∅c0 3437 𝒫 cpw 3593 Tr wtr 4119 |
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-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-uni 3828 df-tr 4120 |
This theorem is referenced by: ord0 4412 ordom 4627 |
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