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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 5024 | . 2 ⊢ Rel ◡∅ | |
2 | rel0 4769 | . 2 ⊢ Rel ∅ | |
3 | vex 2755 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | vex 2755 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelcnv 4827 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡∅ ↔ 〈𝑦, 𝑥〉 ∈ ∅) |
6 | noel 3441 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
7 | noel 3441 | . . . 4 ⊢ ¬ 〈𝑦, 𝑥〉 ∈ ∅ | |
8 | 6, 7 | 2false 702 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ ↔ 〈𝑦, 𝑥〉 ∈ ∅) |
9 | 5, 8 | bitr4i 187 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡∅ ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
10 | 1, 2, 9 | eqrelriiv 4738 | 1 ⊢ ◡∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 ∅c0 3437 〈cop 3610 ◡ccnv 4643 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 |
This theorem is referenced by: xp0 5066 cnveq0 5103 co01 5161 f10 5514 f1o00 5515 tpos0 6300 |
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