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Mirrors > Home > MPE Home > Th. List > tpssi | Structured version Visualization version GIF version |
Description: A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Ref | Expression |
---|---|
tpssi | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4130 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prssi 4293 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) | |
3 | 2 | 3adant3 1074 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) |
4 | snssi 4280 | . . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷) | |
5 | 4 | 3ad2ant3 1077 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷) |
6 | 3, 5 | unssd 3751 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) |
7 | 1, 6 | syl5eqss 3612 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 {csn 4125 {cpr 4127 {ctp 4129 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 df-tp 4130 |
This theorem is referenced by: lcmftp 15187 trgcgrg 25210 2trllemG 26088 sgnclre 29928 signstf 29969 fourierdlem46 39045 fourierdlem102 39101 fourierdlem114 39113 etransclem48 39175 |
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