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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 6096 | . 2 ⊢ Rel (𝐴 ∘ ∅) | |
2 | rel0 5671 | . 2 ⊢ Rel ∅ | |
3 | br0 5114 | . . . . . 6 ⊢ ¬ 𝑥∅𝑧 | |
4 | 3 | intnanr 490 | . . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
5 | 4 | nex 1797 | . . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
6 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | opelco 5741 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)) |
9 | 5, 8 | mtbir 325 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) |
10 | noel 4295 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
11 | 9, 10 | 2false 378 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
12 | 1, 2, 11 | eqrelriiv 5662 | 1 ⊢ (𝐴 ∘ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∅c0 4290 〈cop 4572 class class class wbr 5065 ∘ ccom 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-co 5563 |
This theorem is referenced by: co01 6113 gsumwmhm 18009 frmdgsum 18026 frmdup1 18028 efginvrel2 18852 0frgp 18904 evl1fval 20490 utop2nei 22858 tngds 23256 tocycf 30759 tocyc01 30760 mrsub0 32763 dfpo2 32991 cononrel1 39952 |
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