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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 5084 | . 2 ⊢ Rel (𝐴 ∘ ∅) | |
2 | rel0 4711 | . 2 ⊢ Rel ∅ | |
3 | noel 3398 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑧〉 ∈ ∅ | |
4 | df-br 3966 | . . . . . . 7 ⊢ (𝑥∅𝑧 ↔ 〈𝑥, 𝑧〉 ∈ ∅) | |
5 | 3, 4 | mtbir 661 | . . . . . 6 ⊢ ¬ 𝑥∅𝑧 |
6 | 5 | intnanr 916 | . . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
7 | 6 | nex 1480 | . . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
8 | vex 2715 | . . . . 5 ⊢ 𝑥 ∈ V | |
9 | vex 2715 | . . . . 5 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | opelco 4758 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)) |
11 | 7, 10 | mtbir 661 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) |
12 | noel 3398 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
13 | 11, 12 | 2false 691 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
14 | 1, 2, 13 | eqrelriiv 4680 | 1 ⊢ (𝐴 ∘ ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∃wex 1472 ∈ wcel 2128 ∅c0 3394 〈cop 3563 class class class wbr 3965 ∘ ccom 4590 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-xp 4592 df-rel 4593 df-co 4595 |
This theorem is referenced by: co01 5100 |
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