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

Theorem oe0lem 8127
Description: A helper lemma for oe0 8136 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 413 . . 3 (𝜑 → (𝐴 = ∅ → 𝜓))
32adantl 482 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓))
4 on0eln0 6239 . . . 4 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
54adantr 481 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6 oe0lem.2 . . . 4 (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)
76ex 413 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝜓))
85, 7sylbird 261 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓))
93, 8pm2.61dne 3100 1 ((𝐴 ∈ On ∧ 𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  c0 4288  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  oe0  8136  oev2  8137  oesuclem  8139  oecl  8151  odi  8194  oewordri  8207  oelim2  8210  oeoa  8212  oeoe  8214
  Copyright terms: Public domain W3C validator