|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 4189 | . 2 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) | |
| 2 | df-3an 1088 | . . 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 4784 | . . . 4 ⊢ (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷) | 
| 8 | 5, 7 | anbi12i 628 | . . 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 4630 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
| 11 | 10 | sseq1i 4011 | . 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 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ⊆ wss 3950 {csn 4625 {cpr 4627 {ctp 4629 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-ss 3967 df-sn 4626 df-pr 4628 df-tp 4630 | 
| This theorem is referenced by: 1cubr 26886 konigsberglem4 30275 rabren3dioph 42831 fourierdlem102 46228 fourierdlem114 46240 nnsum4primesodd 47788 nnsum4primesoddALTV 47789 | 
| Copyright terms: Public domain | W3C validator |