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Mirrors > Home > ILE Home > Th. List > co02 | GIF version |
Description: Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
co02 | ⊢ (𝐴 ∘ ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5037 | . 2 ⊢ Rel (𝐴 ∘ ∅) | |
2 | rel0 4664 | . 2 ⊢ Rel ∅ | |
3 | noel 3367 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑧〉 ∈ ∅ | |
4 | df-br 3930 | . . . . . . 7 ⊢ (𝑥∅𝑧 ↔ 〈𝑥, 𝑧〉 ∈ ∅) | |
5 | 3, 4 | mtbir 660 | . . . . . 6 ⊢ ¬ 𝑥∅𝑧 |
6 | 5 | intnanr 915 | . . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
7 | 6 | nex 1476 | . . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
8 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
9 | vex 2689 | . . . . 5 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | opelco 4711 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)) |
11 | 7, 10 | mtbir 660 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) |
12 | noel 3367 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
13 | 11, 12 | 2false 690 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
14 | 1, 2, 13 | eqrelriiv 4633 | 1 ⊢ (𝐴 ∘ ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∅c0 3363 〈cop 3530 class class class wbr 3929 ∘ ccom 4543 |
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-fal 1337 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-co 4548 |
This theorem is referenced by: co01 5053 |
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