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Theorem onfrALTlem3 43305
Description: Lemma for onfrALT 43310. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
Distinct variable groups:   𝑦,𝑎   𝑥,𝑦

Proof of Theorem onfrALTlem3
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 ssid 4005 . . 3 (𝑎𝑥) ⊆ (𝑎𝑥)
2 simpr 486 . . . . 5 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅)
32a1i 11 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅))
4 df-ne 2942 . . . 4 ((𝑎𝑥) ≠ ∅ ↔ ¬ (𝑎𝑥) = ∅)
53, 4syl6ibr 252 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (𝑎𝑥) ≠ ∅))
6 pm3.2 471 . . 3 ((𝑎𝑥) ⊆ (𝑎𝑥) → ((𝑎𝑥) ≠ ∅ → ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)))
71, 5, 6mpsylsyld 69 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)))
8 vex 3479 . . . . 5 𝑥 ∈ V
98inex2 5319 . . . 4 (𝑎𝑥) ∈ V
10 inss2 4230 . . . . . . 7 (𝑎𝑥) ⊆ 𝑥
11 simpl 484 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
12 simpl 484 . . . . . . . . . 10 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥𝑎)
13 ssel 3976 . . . . . . . . . 10 (𝑎 ⊆ On → (𝑥𝑎𝑥 ∈ On))
1411, 12, 13syl2im 40 . . . . . . . . 9 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥 ∈ On))
15 eloni 6375 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
1614, 15syl6 35 . . . . . . . 8 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Ord 𝑥))
17 ordwe 6378 . . . . . . . 8 (Ord 𝑥 → E We 𝑥)
1816, 17syl6 35 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We 𝑥))
19 wess 5664 . . . . . . 7 ((𝑎𝑥) ⊆ 𝑥 → ( E We 𝑥 → E We (𝑎𝑥)))
2010, 18, 19mpsylsyld 69 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We (𝑎𝑥)))
21 wefr 5667 . . . . . 6 ( E We (𝑎𝑥) → E Fr (𝑎𝑥))
2220, 21syl6 35 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E Fr (𝑎𝑥)))
23 dfepfr 5662 . . . . 5 ( E Fr (𝑎𝑥) ↔ ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅))
2422, 23imbitrdi 250 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅)))
25 spsbc 3791 . . . 4 ((𝑎𝑥) ∈ V → (∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) → [(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅)))
269, 24, 25mpsylsyld 69 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → [(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅)))
27 onfrALTlem5 43303 . . 3 ([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
2826, 27imbitrdi 250 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅)))
297, 28mpdd 43 1 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  wne 2941  wrex 3071  Vcvv 3475  [wsbc 3778  cin 3948  wss 3949  c0 4323   E cep 5580   Fr wfr 5629   We wwe 5631  Ord word 6364  Oncon0 6365
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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369
This theorem is referenced by:  onfrALTlem2  43307
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