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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 5045 | . 2 ⊢ Rel (𝐴 ∘ ∅) | |
2 | rel0 4672 | . 2 ⊢ Rel ∅ | |
3 | noel 3372 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑧〉 ∈ ∅ | |
4 | df-br 3938 | . . . . . . 7 ⊢ (𝑥∅𝑧 ↔ 〈𝑥, 𝑧〉 ∈ ∅) | |
5 | 3, 4 | mtbir 661 | . . . . . 6 ⊢ ¬ 𝑥∅𝑧 |
6 | 5 | intnanr 916 | . . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
7 | 6 | nex 1477 | . . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
8 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
9 | vex 2692 | . . . . 5 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | opelco 4719 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)) |
11 | 7, 10 | mtbir 661 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) |
12 | noel 3372 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
13 | 11, 12 | 2false 691 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
14 | 1, 2, 13 | eqrelriiv 4641 | 1 ⊢ (𝐴 ∘ ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∅c0 3368 〈cop 3535 class class class wbr 3937 ∘ ccom 4551 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-co 4556 |
This theorem is referenced by: co01 5061 |
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