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

Theorem sorpssi 6985
Description: Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpssi (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))

Proof of Theorem sorpssi
StepHypRef Expression
1 solin 5087 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵))
2 elex 3243 . . . . . 6 (𝐶𝐴𝐶 ∈ V)
32ad2antll 765 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ V)
4 brrpssg 6981 . . . . 5 (𝐶 ∈ V → (𝐵 [] 𝐶𝐵𝐶))
53, 4syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵𝐶))
6 biidd 252 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶𝐵 = 𝐶))
7 elex 3243 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
87ad2antrl 764 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ V)
9 brrpssg 6981 . . . . 5 (𝐵 ∈ V → (𝐶 [] 𝐵𝐶𝐵))
108, 9syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶 [] 𝐵𝐶𝐵))
115, 6, 103orbi123d 1438 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵)))
121, 11mpbid 222 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
13 sspsstri 3742 . 2 ((𝐵𝐶𝐶𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
1412, 13sylibr 224 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3o 1053   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  wpss 3608   class class class wbr 4685   Or wor 5063   [] crpss 6978
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  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  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-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-so 5065  df-xp 5149  df-rel 5150  df-rpss 6979
This theorem is referenced by:  sorpssun  6986  sorpssin  6987  sorpssuni  6988  sorpssint  6989  sorpsscmpl  6990  enfin2i  9181  fin1a2lem9  9268  fin1a2lem10  9269  fin1a2lem11  9270  fin1a2lem13  9272
  Copyright terms: Public domain W3C validator