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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 4875 | . 2 ⊢ Rel ◡∅ | |
2 | rel0 4624 | . 2 ⊢ Rel ∅ | |
3 | vex 2660 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | vex 2660 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelcnv 4681 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡∅ ↔ 〈𝑦, 𝑥〉 ∈ ∅) |
6 | noel 3333 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
7 | noel 3333 | . . . 4 ⊢ ¬ 〈𝑦, 𝑥〉 ∈ ∅ | |
8 | 6, 7 | 2false 673 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ ↔ 〈𝑦, 𝑥〉 ∈ ∅) |
9 | 5, 8 | bitr4i 186 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡∅ ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
10 | 1, 2, 9 | eqrelriiv 4593 | 1 ⊢ ◡∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 ∈ wcel 1463 ∅c0 3329 〈cop 3496 ◡ccnv 4498 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-opab 3950 df-xp 4505 df-rel 4506 df-cnv 4507 |
This theorem is referenced by: xp0 4916 cnveq0 4953 co01 5011 f10 5357 f1o00 5358 tpos0 6125 |
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