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Theorem onfrALTlem3VD 43648
Description: Virtual deduction proof of onfrALTlem3 43305. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem3 43305 is onfrALTlem3VD 43648 without virtual deductions and was automatically derived from onfrALTlem3VD 43648.
1:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   )
3:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥𝑎   )
4:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 On   )
5:3,4: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥 ∈ On   )
6:5: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   Ord 𝑥   )
7:6: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E We 𝑥   )
8:: (𝑎𝑥) ⊆ 𝑥
9:7,8: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E We (𝑎𝑥)   )
10:9: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E Fr (𝑎𝑥)   )
11:10: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅)   )
12:: 𝑥 ∈ V
13:12,8: (𝑎𝑥) ∈ V
14:13,11: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   [(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅)   )
15:: ([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏(𝑏𝑦) = ∅) ↔ (((𝑎 𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)( (𝑎𝑥) ∩ 𝑦) = ∅))
16:14,15: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ ( 𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅)   )
17:: (𝑎𝑥) ⊆ (𝑎𝑥)
18:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   ¬ (𝑎𝑥) = ∅   )
19:18: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (𝑎𝑥) ≠ ∅   )
20:17,19: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎 𝑥) ≠ ∅)   )
qed:16,20: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦 ) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem3VD (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅   )
Distinct variable groups:   𝑦,𝑎   𝑥,𝑦

Proof of Theorem onfrALTlem3VD
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 vex 3479 . . . . 5 𝑥 ∈ V
2 inss2 4230 . . . . 5 (𝑎𝑥) ⊆ 𝑥
31, 2ssexi 5323 . . . 4 (𝑎𝑥) ∈ V
4 idn2 43374 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   )
5 simpl 484 . . . . . . . . . . 11 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥𝑎)
64, 5e2 43392 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥𝑎   )
7 idn1 43335 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
8 simpl 484 . . . . . . . . . . 11 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
97, 8e1a 43388 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ⊆ On   )
10 ssel 3976 . . . . . . . . . . 11 (𝑎 ⊆ On → (𝑥𝑎𝑥 ∈ On))
1110com12 32 . . . . . . . . . 10 (𝑥𝑎 → (𝑎 ⊆ On → 𝑥 ∈ On))
126, 9, 11e21 43491 . . . . . . . . 9 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑥 ∈ On   )
13 eloni 6375 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
1412, 13e2 43392 . . . . . . . 8 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   Ord 𝑥   )
15 ordwe 6378 . . . . . . . 8 (Ord 𝑥 → E We 𝑥)
1614, 15e2 43392 . . . . . . 7 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E We 𝑥   )
17 wess 5664 . . . . . . . 8 ((𝑎𝑥) ⊆ 𝑥 → ( E We 𝑥 → E We (𝑎𝑥)))
1817com12 32 . . . . . . 7 ( E We 𝑥 → ((𝑎𝑥) ⊆ 𝑥 → E We (𝑎𝑥)))
1916, 2, 18e20 43488 . . . . . 6 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E We (𝑎𝑥)   )
20 wefr 5667 . . . . . 6 ( E We (𝑎𝑥) → E Fr (𝑎𝑥))
2119, 20e2 43392 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    E Fr (𝑎𝑥)   )
22 dfepfr 5662 . . . . . 6 ( E Fr (𝑎𝑥) ↔ ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅))
2322biimpi 215 . . . . 5 ( E Fr (𝑎𝑥) → ∀𝑏((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅))
2421, 23e2 43392 . . . 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) = ∅)))
263, 24, 25e02 43458 . . 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, 27e2bi 43393 . 2 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅)   )
29 ssid 4005 . . 3 (𝑎𝑥) ⊆ (𝑎𝑥)
30 simpr 486 . . . . 5 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ¬ (𝑎𝑥) = ∅)
314, 30e2 43392 . . . 4 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶    ¬ (𝑎𝑥) = ∅   )
32 df-ne 2942 . . . . 5 ((𝑎𝑥) ≠ ∅ ↔ ¬ (𝑎𝑥) = ∅)
3332biimpri 227 . . . 4 (¬ (𝑎𝑥) = ∅ → (𝑎𝑥) ≠ ∅)
3431, 33e2 43392 . . 3 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   (𝑎𝑥) ≠ ∅   )
35 pm3.2 471 . . 3 ((𝑎𝑥) ⊆ (𝑎𝑥) → ((𝑎𝑥) ≠ ∅ → ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)))
3629, 34, 35e02 43458 . 2 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   ((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅)   )
37 id 22 . 2 ((((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅) → (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
3828, 36, 37e22 43432 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  (   wvd2 43338
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  df-vd1 43331  df-vd2 43339
This theorem is referenced by:  onfrALTlem2VD  43650
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