MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brrpssg Structured version   Visualization version   GIF version

Theorem brrpssg 7166
Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
brrpssg (𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))

Proof of Theorem brrpssg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3405 . . 3 (𝐵𝑉𝐵 ∈ V)
2 relrpss 7165 . . . 4 Rel []
32brrelexi 5355 . . 3 (𝐴 [] 𝐵𝐴 ∈ V)
41, 3anim12i 602 . 2 ((𝐵𝑉𝐴 [] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
51adantr 468 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐵 ∈ V)
6 pssss 3897 . . . 4 (𝐴𝐵𝐴𝐵)
7 ssexg 4996 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
86, 1, 7syl2anr 586 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐴 ∈ V)
95, 8jca 503 . 2 ((𝐵𝑉𝐴𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
10 psseq1 3889 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 psseq2 3890 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-rpss 7164 . . . 4 [] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5186 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
1413ancoms 448 . 2 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
154, 9, 14pm5.21nd 827 1 (𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2158  Vcvv 3390  wss 3766  wpss 3767   class class class wbr 4840   [] crpss 7163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pr 5093
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4841  df-opab 4903  df-xp 5314  df-rel 5315  df-rpss 7164
This theorem is referenced by:  brrpss  7167  sorpssi  7170
  Copyright terms: Public domain W3C validator