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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 5633 | . 2 ⊢ Rel (𝐴 ∘ ∅) | |
2 | rel0 5243 | . 2 ⊢ Rel ∅ | |
3 | br0 4701 | . . . . . 6 ⊢ ¬ 𝑥∅𝑧 | |
4 | 3 | intnanr 961 | . . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
5 | 4 | nex 1731 | . . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
6 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | opelco 5293 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)) |
9 | 5, 8 | mtbir 313 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) |
10 | noel 3919 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
11 | 9, 10 | 2false 365 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
12 | 1, 2, 11 | eqrelriiv 5214 | 1 ⊢ (𝐴 ∘ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∅c0 3915 〈cop 4183 class class class wbr 4653 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-co 5123 |
This theorem is referenced by: co01 5650 gsumwmhm 17382 frmdgsum 17399 frmdup1 17401 efginvrel2 18140 0frgp 18192 evl1fval 19692 utop2nei 22054 tngds 22452 mrsub0 31413 dfpo2 31645 cononrel1 37900 |
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