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Theorem oe0lem 7762
Description: A helper lemma for oe0 7771 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 449 . . 3 (𝜑 → (𝐴 = ∅ → 𝜓))
32adantl 473 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓))
4 on0eln0 5941 . . . 4 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
54adantr 472 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6 oe0lem.2 . . . 4 (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)
76ex 449 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝜓))
85, 7sylbird 250 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓))
93, 8pm2.61dne 3018 1 ((𝐴 ∈ On ∧ 𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  c0 4058  Oncon0 5884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888
This theorem is referenced by:  oe0  7771  oev2  7772  oesuclem  7774  oecl  7786  odi  7828  oewordri  7841  oelim2  7844  oeoa  7846  oeoe  7848
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