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Mirrors > Home > MPE Home > Th. List > tpss | Structured version Visualization version GIF version |
Description: An unordered triple of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
tpss.1 | ⊢ 𝐴 ∈ V |
tpss.2 | ⊢ 𝐵 ∈ V |
tpss.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
tpss | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss 4180 | . 2 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) | |
2 | df-3an 1087 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷)) | |
3 | tpss.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | tpss.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | prss 4819 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷) |
6 | tpss.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
7 | 6 | snss 4785 | . . . 4 ⊢ (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷) |
8 | 5, 7 | anbi12i 627 | . . 3 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷)) |
9 | 2, 8 | bitri 275 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷)) |
10 | df-tp 4629 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
11 | 10 | sseq1i 4006 | . 2 ⊢ ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) |
12 | 1, 9, 11 | 3bitr4i 303 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2099 Vcvv 3470 ∪ cun 3943 ⊆ wss 3945 {csn 4624 {cpr 4626 {ctp 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-un 3950 df-in 3952 df-ss 3962 df-sn 4625 df-pr 4627 df-tp 4629 |
This theorem is referenced by: 1cubr 26767 konigsberglem4 30058 rabren3dioph 42229 fourierdlem102 45590 fourierdlem114 45602 nnsum4primesodd 47130 nnsum4primesoddALTV 47131 |
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