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Theorem oe0lem 7834
Description: A helper lemma for oe0 7843 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 402 . . 3 (𝜑 → (𝐴 = ∅ → 𝜓))
32adantl 474 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 = ∅ → 𝜓))
4 on0eln0 5997 . . . 4 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
54adantr 473 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝐴 ≠ ∅))
6 oe0lem.2 . . . 4 (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)
76ex 402 . . 3 ((𝐴 ∈ On ∧ 𝜑) → (∅ ∈ 𝐴𝜓))
85, 7sylbird 252 . 2 ((𝐴 ∈ On ∧ 𝜑) → (𝐴 ≠ ∅ → 𝜓))
93, 8pm2.61dne 3058 1 ((𝐴 ∈ On ∧ 𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wne 2972  c0 4116  Oncon0 5942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-ord 5945  df-on 5946
This theorem is referenced by:  oe0  7843  oev2  7844  oesuclem  7846  oecl  7858  odi  7900  oewordri  7913  oelim2  7916  oeoa  7918  oeoe  7920
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