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Theorem oe0lem 7538
Description: A helper lemma for oe0 7547 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 450 . . 3 (𝜑 → (𝐴 = ∅ → 𝜓))
32adantl 482 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓))
4 on0eln0 5739 . . . 4 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
54adantr 481 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6 oe0lem.2 . . . 4 (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)
76ex 450 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝜓))
85, 7sylbird 250 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓))
93, 8pm2.61dne 2876 1 ((𝐴 ∈ On ∧ 𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  c0 3891  Oncon0 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686
This theorem is referenced by:  oe0  7547  oev2  7548  oesuclem  7550  oecl  7562  odi  7604  oewordri  7617  oelim2  7620  oeoa  7622  oeoe  7624
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