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Theorem onfrALTlem1 42922
Description: Lemma for onfrALT 42923. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem1 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
Distinct variable group:   𝑥,𝑎,𝑦

Proof of Theorem onfrALTlem1
StepHypRef Expression
1 19.8a 2175 . . . . 5 ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅))
21a1i 11 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅)))
3 cbvexsv 42921 . . . 4 (∃𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅))
42, 3syl6ib 251 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅)))
5 sbsbc 3747 . . . . 5 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ [𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅))
6 onfrALTlem4 42917 . . . . 5 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
75, 6bitri 275 . . . 4 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
87exbii 1851 . . 3 (∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
94, 8syl6ib 251 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)))
10 df-rex 3071 . 2 (∃𝑦𝑎 (𝑎𝑦) = ∅ ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
119, 10syl6ibr 252 1 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  [wsb 2068  wne 2940  wrex 3070  [wsbc 3743  cin 3913  wss 3914  c0 4286  Oncon0 6321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-in 3921  df-nul 4287
This theorem is referenced by:  onfrALT  42923
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