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Mirrors > Home > MPE Home > Th. List > prssd | Structured version Visualization version GIF version |
Description: Deduction version of prssi 4293: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
prssd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
prssd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
prssd | ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | prssd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
3 | prssi 4293 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ⊆ 𝐶) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 {cpr 4127 |
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 |
This theorem is referenced by: nehash2 13113 dchrisum0re 25002 uhgrstrrepelem 25744 upgrex 25759 upgr1e 25779 poimirlem9 32588 clsk1indlem4 37362 clsk1indlem1 37363 meadjun 39355 uspgr1e 40470 eupth2lems 41406 |
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