Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem11 Structured version   Visualization version   GIF version

Theorem prtlem11 34470
Description: Lemma for prter2 34485. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
prtlem11 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))

Proof of Theorem prtlem11
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 risset 3091 . . . 4 (𝐶𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐶)
2 r19.41v 3118 . . . . 5 (∃𝑥𝐴 (𝑥 = 𝐶𝐵 = [𝐶] ) ↔ (∃𝑥𝐴 𝑥 = 𝐶𝐵 = [𝐶] ))
3 eceq1 7825 . . . . . . 7 (𝑥 = 𝐶 → [𝑥] = [𝐶] )
4 eqtr3 2672 . . . . . . . 8 (([𝑥] = [𝐶] 𝐵 = [𝐶] ) → [𝑥] = 𝐵)
54eqcomd 2657 . . . . . . 7 (([𝑥] = [𝐶] 𝐵 = [𝐶] ) → 𝐵 = [𝑥] )
63, 5sylan 487 . . . . . 6 ((𝑥 = 𝐶𝐵 = [𝐶] ) → 𝐵 = [𝑥] )
76reximi 3040 . . . . 5 (∃𝑥𝐴 (𝑥 = 𝐶𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
82, 7sylbir 225 . . . 4 ((∃𝑥𝐴 𝑥 = 𝐶𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
91, 8sylanb 488 . . 3 ((𝐶𝐴𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
10 elqsg 7841 . . 3 (𝐵𝐷 → (𝐵 ∈ (𝐴 / ) ↔ ∃𝑥𝐴 𝐵 = [𝑥] ))
119, 10syl5ibr 236 . 2 (𝐵𝐷 → ((𝐶𝐴𝐵 = [𝐶] ) → 𝐵 ∈ (𝐴 / )))
1211expd 451 1 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wrex 2942  [cec 7785   / cqs 7786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ec 7789  df-qs 7793
This theorem is referenced by:  prter2  34485
  Copyright terms: Public domain W3C validator