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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 3602 | . 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 1439 ⊆ wss 3002 {cpr 3453 |
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 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2624 df-un 3006 df-in 3008 df-ss 3015 df-sn 3458 df-pr 3459 |
This theorem is referenced by: tpssi 3611 prelpwi 4052 onun2 4322 onintexmid 4403 nnregexmid 4449 en2eqpr 6679 m1expcl2 10040 m1expcl 10041 minmax 10724 1idssfct 11438 unopn 11767 bdop 12070 |
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