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Theorem onfrALTlem3 44542
Description: Lemma for onfrALT 44547. (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 4018 . . 3 (𝑎𝑥) ⊆ (𝑎𝑥)
2 simpr 484 . . . . 5 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅)
32a1i 11 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅))
4 df-ne 2939 . . . 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 3482 . . . . 5 𝑥 ∈ V
98inex2 5324 . . . 4 (𝑎𝑥) ∈ V
10 inss2 4246 . . . . . . 7 (𝑎𝑥) ⊆ 𝑥
11 simpl 482 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
12 simpl 482 . . . . . . . . . 10 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥𝑎)
13 ssel 3989 . . . . . . . . . 10 (𝑎 ⊆ On → (𝑥𝑎𝑥 ∈ On))
1411, 12, 13syl2im 40 . . . . . . . . 9 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥 ∈ On))
15 eloni 6396 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
1614, 15syl6 35 . . . . . . . 8 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Ord 𝑥))
17 ordwe 6399 . . . . . . . 8 (Ord 𝑥 → E We 𝑥)
1816, 17syl6 35 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We 𝑥))
19 wess 5675 . . . . . . 7 ((𝑎𝑥) ⊆ 𝑥 → ( E We 𝑥 → E We (𝑎𝑥)))
2010, 18, 19mpsylsyld 69 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E We (𝑎𝑥)))
21 wefr 5679 . . . . . 6 ( E We (𝑎𝑥) → E Fr (𝑎𝑥))
2220, 21syl6 35 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → E Fr (𝑎𝑥)))
23 dfepfr 5673 . . . . 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 3804 . . . 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 44540 . . 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 1535   = wceq 1537  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  [wsbc 3791  cin 3962  wss 3963  c0 4339   E cep 5588   Fr wfr 5638   We wwe 5640  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onfrALTlem2  44544
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