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Theorem ssintub 3849
Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
Assertion
Ref Expression
ssintub 𝐴 {𝑥𝐵𝐴𝑥}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssintub
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssint 3847 . 2 (𝐴 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐴𝑦)
2 sseq2 3171 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32elrab 2886 . . 3 (𝑦 ∈ {𝑥𝐵𝐴𝑥} ↔ (𝑦𝐵𝐴𝑦))
43simprbi 273 . 2 (𝑦 ∈ {𝑥𝐵𝐴𝑥} → 𝐴𝑦)
51, 4mprgbir 2528 1 𝐴 {𝑥𝐵𝐴𝑥}
Colors of variables: wff set class
Syntax hints:  wcel 2141  {crab 2452  wss 3121   cint 3831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-int 3832
This theorem is referenced by:  intmin  3851  sscls  12914
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