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Theorem unimax 3869
Description: Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
Assertion
Ref Expression
unimax (𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unimax
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssid 3199 . . 3 𝐴𝐴
2 sseq1 3202 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
32elrab3 2917 . . 3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝑥𝐴} ↔ 𝐴𝐴))
41, 3mpbiri 168 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝐵𝑥𝐴})
5 sseq1 3202 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65elrab 2916 . . . 4 (𝑦 ∈ {𝑥𝐵𝑥𝐴} ↔ (𝑦𝐵𝑦𝐴))
76simprbi 275 . . 3 (𝑦 ∈ {𝑥𝐵𝑥𝐴} → 𝑦𝐴)
87rgen 2547 . 2 𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴
9 ssunieq 3868 . . 3 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → 𝐴 = {𝑥𝐵𝑥𝐴})
109eqcomd 2199 . 2 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → {𝑥𝐵𝑥𝐴} = 𝐴)
114, 8, 10sylancl 413 1 (𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2164  wral 2472  {crab 2476  wss 3153   cuni 3835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836
This theorem is referenced by:  onuniss2  4544  lssuni  13859
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