Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cnv0 | GIF version |
Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) |
Ref | Expression |
---|---|
cnv0 | ⊢ ◡∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4917 | . 2 ⊢ Rel ◡∅ | |
2 | rel0 4664 | . 2 ⊢ Rel ∅ | |
3 | vex 2689 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | vex 2689 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelcnv 4721 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡∅ ↔ 〈𝑦, 𝑥〉 ∈ ∅) |
6 | noel 3367 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
7 | noel 3367 | . . . 4 ⊢ ¬ 〈𝑦, 𝑥〉 ∈ ∅ | |
8 | 6, 7 | 2false 690 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ ↔ 〈𝑦, 𝑥〉 ∈ ∅) |
9 | 5, 8 | bitr4i 186 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡∅ ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
10 | 1, 2, 9 | eqrelriiv 4633 | 1 ⊢ ◡∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ∅c0 3363 〈cop 3530 ◡ccnv 4538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 |
This theorem is referenced by: xp0 4958 cnveq0 4995 co01 5053 f10 5401 f1o00 5402 tpos0 6171 |
Copyright terms: Public domain | W3C validator |