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Mirrors > Home > ILE Home > Th. List > prssi | GIF version |
Description: A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
prssi | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssg 3616 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)) | |
2 | 1 | ibi 175 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1445 ⊆ wss 3013 {cpr 3467 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-pr 3473 |
This theorem is referenced by: tpssi 3625 prelpwi 4065 onun2 4335 onintexmid 4416 nnregexmid 4462 en2eqpr 6703 m1expcl2 10108 m1expcl 10109 minmax 10792 xrminmax 10824 1idssfct 11540 unopn 11872 bdop 12490 |
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