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Mirrors > Home > ILE Home > Th. List > tpss | GIF version |
Description: A triplet 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 3216 | . 2 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) | |
2 | df-3an 947 | . . 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 3642 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷) |
6 | tpss.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
7 | 6 | snss 3615 | . . . 4 ⊢ (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷) |
8 | 5, 7 | anbi12i 453 | . . 3 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷)) |
9 | 2, 8 | bitri 183 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷)) |
10 | df-tp 3501 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
11 | 10 | sseq1i 3089 | . 2 ⊢ ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) |
12 | 1, 9, 11 | 3bitr4i 211 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 945 ∈ wcel 1463 Vcvv 2657 ∪ cun 3035 ⊆ wss 3037 {csn 3493 {cpr 3494 {ctp 3495 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-tp 3501 |
This theorem is referenced by: (None) |
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