![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dmsnopss | Structured version Visualization version GIF version |
Description: The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on 𝐵). (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopss | ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmsnopg 6212 | . . 3 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴}) | |
2 | eqimss 4040 | . . 3 ⊢ (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}) |
4 | opprc2 4898 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅) | |
5 | 4 | sneqd 4640 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅}) |
6 | 5 | dmeqd 5905 | . . . 4 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅}) |
7 | dmsn0 6208 | . . . 4 ⊢ dom {∅} = ∅ | |
8 | 6, 7 | eqtrdi 2788 | . . 3 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅) |
9 | 0ss 4396 | . . 3 ⊢ ∅ ⊆ {𝐴} | |
10 | 8, 9 | eqsstrdi 4036 | . 2 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}) |
11 | 3, 10 | pm2.61i 182 | 1 ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 {csn 4628 ⟨cop 4634 dom cdm 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-dm 5686 |
This theorem is referenced by: snopsuppss 8163 strle1 17090 setsres 17110 setscom 17112 setsid 17140 ex-res 29691 bj-fununsn1 36129 mapfzcons1 41445 |
Copyright terms: Public domain | W3C validator |