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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 4564 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prssi 4746 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) | |
3 | 2 | 3adant3 1127 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) |
4 | snssi 4733 | . . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷) | |
5 | 4 | 3ad2ant3 1130 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷) |
6 | 3, 5 | unssd 4160 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) |
7 | 1, 6 | eqsstrid 4013 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 ∈ wcel 2108 ∪ cun 3932 ⊆ wss 3934 {csn 4559 {cpr 4561 {ctp 4563 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-v 3495 df-un 3939 df-in 3941 df-ss 3950 df-sn 4560 df-pr 4562 df-tp 4564 |
This theorem is referenced by: lcmftp 15972 trgcgrg 26293 cyc3co2 30775 sgnclre 31790 signstf 31829 limsupequzlem 41993 fourierdlem46 42428 fourierdlem102 42484 fourierdlem114 42496 etransclem48 42558 |
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