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Theorem onfrALTlem3 44569
Description: Lemma for onfrALT 44574. (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 484 . . . . 5 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅)
32a1i 11 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅))
4 df-ne 2940 . . . 4 ((𝑎𝑥) ≠ ∅ ↔ ¬ (𝑎𝑥) = ∅)
53, 4imbitrrdi 252 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (𝑎𝑥) ≠ ∅))
6 pm3.2 469 . . 3 ((𝑎𝑥) ⊆ (𝑎𝑥) → ((𝑎𝑥) ≠ ∅ → ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)))
71, 5, 6mpsylsyld 69 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)))
8 vex 3483 . . . . 5 𝑥 ∈ V
98inex2 5317 . . . 4 (𝑎𝑥) ∈ V
10 inss2 4237 . . . . . . 7 (𝑎𝑥) ⊆ 𝑥
11 simpl 482 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
12 simpl 482 . . . . . . . . . 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 6393 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
1614, 15syl6 35 . . . . . . . 8 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Ord 𝑥))
17 ordwe 6396 . . . . . . . 8 (Ord 𝑥 → E We 𝑥)
1816, 17syl6 35 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We 𝑥))
19 wess 5670 . . . . . . 7 ((𝑎𝑥) ⊆ 𝑥 → ( E We 𝑥 → E We (𝑎𝑥)))
2010, 18, 19mpsylsyld 69 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We (𝑎𝑥)))
21 wefr 5674 . . . . . 6 ( E We (𝑎𝑥) → E Fr (𝑎𝑥))
2220, 21syl6 35 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E Fr (𝑎𝑥)))
23 dfepfr 5668 . . . . 5 ( E Fr (𝑎𝑥) ↔ ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅))
2422, 23imbitrdi 251 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅)))
25 spsbc 3800 . . . 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 44567 . . 3 ([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
2826, 27imbitrdi 251 . 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 395  wal 1537   = wceq 1539  wcel 2107  wne 2939  wrex 3069  Vcvv 3479  [wsbc 3787  cin 3949  wss 3950  c0 4332   E cep 5582   Fr wfr 5633   We wwe 5635  Ord word 6382  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  onfrALTlem2  44571
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