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Theorem unimax 3770
 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 3117 . . 3 𝐴𝐴
2 sseq1 3120 . . . 4 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
32elrab3 2841 . . 3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝑥𝐴} ↔ 𝐴𝐴))
41, 3mpbiri 167 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝐵𝑥𝐴})
5 sseq1 3120 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65elrab 2840 . . . 4 (𝑦 ∈ {𝑥𝐵𝑥𝐴} ↔ (𝑦𝐵𝑦𝐴))
76simprbi 273 . . 3 (𝑦 ∈ {𝑥𝐵𝑥𝐴} → 𝑦𝐴)
87rgen 2485 . 2 𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴
9 ssunieq 3769 . . 3 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → 𝐴 = {𝑥𝐵𝑥𝐴})
109eqcomd 2145 . 2 ((𝐴 ∈ {𝑥𝐵𝑥𝐴} ∧ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴) → {𝑥𝐵𝑥𝐴} = 𝐴)
114, 8, 10sylancl 409 1 (𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416  {crab 2420   ⊆ wss 3071  ∪ cuni 3736 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737 This theorem is referenced by:  onuniss2  4428
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