Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5550 | . 2 ⊢ Rel (𝐴 ∘ ∅) | |
2 | rel0 5166 | . 2 ⊢ Rel ∅ | |
3 | br0 4631 | . . . . . 6 ⊢ ¬ 𝑥∅𝑧 | |
4 | 3 | intnanr 952 | . . . . 5 ⊢ ¬ (𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
5 | 4 | nex 1722 | . . . 4 ⊢ ¬ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦) |
6 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | opelco 5215 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥∅𝑧 ∧ 𝑧𝐴𝑦)) |
9 | 5, 8 | mtbir 312 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) |
10 | noel 3878 | . . 3 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
11 | 9, 10 | 2false 364 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ ∅) ↔ 〈𝑥, 𝑦〉 ∈ ∅) |
12 | 1, 2, 11 | eqrelriiv 5137 | 1 ⊢ (𝐴 ∘ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∅c0 3874 〈cop 4131 class class class wbr 4583 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-co 5047 |
This theorem is referenced by: co01 5567 gsumwmhm 17205 frmdgsum 17222 frmdup1 17224 efginvrel2 17963 0frgp 18015 evl1fval 19513 utop2nei 21864 tngds 22262 mrsub0 30667 dfpo2 30898 cononrel1 36919 |
Copyright terms: Public domain | W3C validator |