MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe0lem Structured version   Visualization version   GIF version

Theorem oe0lem 8551
Description: A helper lemma for oe0 8560 and others. (Contributed by NM, 6-Jan-2005.)
Hypotheses
Ref Expression
oe0lem.1 ((𝜑𝐴 = ∅) → 𝜓)
oe0lem.2 (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)
Assertion
Ref Expression
oe0lem ((𝐴 ∈ On ∧ 𝜑) → 𝜓)

Proof of Theorem oe0lem
StepHypRef Expression
1 oe0lem.1 . . . 4 ((𝜑𝐴 = ∅) → 𝜓)
21ex 412 . . 3 (𝜑 → (𝐴 = ∅ → 𝜓))
32adantl 481 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓))
4 on0eln0 6440 . . . 4 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
54adantr 480 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6 oe0lem.2 . . . 4 (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)
76ex 412 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝜓))
85, 7sylbird 260 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓))
93, 8pm2.61dne 3028 1 ((𝐴 ∈ On ∧ 𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  c0 4333  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  oe0  8560  oev2  8561  oesuclem  8563  oecl  8575  odi  8617  oewordri  8630  oelim2  8633  oeoa  8635  oeoe  8637
  Copyright terms: Public domain W3C validator