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| Mirrors > Home > MPE Home > Th. List > co02 | Structured version Visualization version 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 6126 | . 2 ⊢ Rel (𝐴 ∘ ∅) | |
| 2 | rel0 5809 | . 2 ⊢ Rel ∅ | |
| 3 | br0 5192 | . . . . . 6 ⊢ ¬ 𝑥∅𝑧 | |
| 4 | 3 | intnanr 487 | . . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
| 5 | 4 | nex 1800 | . . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
| 6 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 7 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 8 | 6, 7 | opelco 5882 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)) |
| 9 | 5, 8 | mtbir 323 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) |
| 10 | noel 4338 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
| 11 | 9, 10 | 2false 375 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
| 12 | 1, 2, 11 | eqrelriiv 5800 | 1 ⊢ (𝐴 ∘ ∅) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∅c0 4333 〈cop 4632 class class class wbr 5143 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-co 5694 |
| This theorem is referenced by: co01 6281 dfpo2 6316 relexpsucld 15073 gsumwmhm 18858 frmdgsum 18875 frmdup1 18877 efginvrel2 19745 0frgp 19797 evl1fval 22332 utop2nei 24259 tngds 24668 tngdsOLD 24669 tocycf 33137 tocyc01 33138 1arithidom 33565 mrsub0 35521 cononrel1 43607 |
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