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Theorem onfrALTlem3 39253
Description: Lemma for onfrALT 39258. (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 3757 . . 3 (𝑎𝑥) ⊆ (𝑎𝑥)
2 simpr 479 . . . . 5 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅)
32a1i 11 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅))
4 df-ne 2925 . . . 4 ((𝑎𝑥) ≠ ∅ ↔ ¬ (𝑎𝑥) = ∅)
53, 4syl6ibr 242 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (𝑎𝑥) ≠ ∅))
6 pm3.2 462 . . 3 ((𝑎𝑥) ⊆ (𝑎𝑥) → ((𝑎𝑥) ≠ ∅ → ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)))
71, 5, 6mpsylsyld 69 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)))
8 vex 3335 . . . . 5 𝑥 ∈ V
98inex2 4944 . . . 4 (𝑎𝑥) ∈ V
10 inss2 3969 . . . . . . 7 (𝑎𝑥) ⊆ 𝑥
11 simpl 474 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
12 simpl 474 . . . . . . . . . 10 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥𝑎)
13 ssel 3730 . . . . . . . . . 10 (𝑎 ⊆ On → (𝑥𝑎𝑥 ∈ On))
1411, 12, 13syl2im 40 . . . . . . . . 9 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥 ∈ On))
15 eloni 5886 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
1614, 15syl6 35 . . . . . . . 8 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Ord 𝑥))
17 ordwe 5889 . . . . . . . 8 (Ord 𝑥 → E We 𝑥)
1816, 17syl6 35 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We 𝑥))
19 wess 5245 . . . . . . 7 ((𝑎𝑥) ⊆ 𝑥 → ( E We 𝑥 → E We (𝑎𝑥)))
2010, 18, 19mpsylsyld 69 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We (𝑎𝑥)))
21 wefr 5248 . . . . . 6 ( E We (𝑎𝑥) → E Fr (𝑎𝑥))
2220, 21syl6 35 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E Fr (𝑎𝑥)))
23 dfepfr 5243 . . . . 5 ( E Fr (𝑎𝑥) ↔ ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅))
2422, 23syl6ib 241 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅)))
25 spsbc 3581 . . . 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 39251 . . 3 ([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
2826, 27syl6ib 241 . 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 383  wal 1622   = wceq 1624  wcel 2131  wne 2924  wrex 3043  Vcvv 3332  [wsbc 3568  cin 3706  wss 3707  c0 4050   E cep 5170   Fr wfr 5214   We wwe 5216  Ord word 5875  Oncon0 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-ord 5879  df-on 5880
This theorem is referenced by:  onfrALTlem2  39255
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