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Theorem onfrALTVD 43652
Description: Virtual deduction proof of onfrALT 43310. 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. onfrALT 43310 is onfrALTVD 43652 without virtual deductions and was automatically derived from onfrALTVD 43652.
1:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎(𝑎𝑦) = ∅   )
2:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎(𝑎𝑦) = ∅   )
3:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶    (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
4:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶    ((𝑎𝑥) = ∅ → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
5:: ((𝑎𝑥) = ∅ ∨ ¬ (𝑎𝑥) = ∅)
6:5,4,3: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶    𝑦𝑎(𝑎𝑦) = ∅   )
7:6: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥𝑎 → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
8:7: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑥(𝑥 𝑎 → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
9:8: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥𝑥 𝑎 → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
10:: (𝑎 ≠ ∅ ↔ ∃𝑥𝑥𝑎)
11:9,10: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ∅ → ∃𝑦𝑎(𝑎𝑦) = ∅)   )
12:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 On ∧ 𝑎 ≠ ∅)   )
13:12: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎    )
14:13,11: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑦 𝑎(𝑎𝑦) = ∅   )
15:14: ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
16:15: 𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 𝑎(𝑎𝑦) = ∅)
qed:16: E Fr On
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTVD E Fr On

Proof of Theorem onfrALTVD
Dummy variables 𝑥 𝑎 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 43335 . . . . . 6 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2 simpr 486 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
31, 2e1a 43388 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
4 exmid 894 . . . . . . . . . 10 ((𝑎𝑥) = ∅ ∨ ¬ (𝑎𝑥) = ∅)
5 onfrALTlem1VD 43651 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
65in2an 43369 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
7 onfrALTlem2VD 43650 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
87in2an 43369 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
9 pm2.61 191 . . . . . . . . . . 11 (((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
109a1i 11 . . . . . . . . . 10 (((𝑎𝑥) = ∅ ∨ ¬ (𝑎𝑥) = ∅) → (((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
114, 6, 8, 10e022 43402 . . . . . . . . 9 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
1211in2 43366 . . . . . . . 8 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
1312gen11 43377 . . . . . . 7 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
14 19.23v 1946 . . . . . . . 8 (∀𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) ↔ (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
1514biimpi 215 . . . . . . 7 (∀𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) → (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
1613, 15e1a 43388 . . . . . 6 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
17 n0 4347 . . . . . 6 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
18 imbi1 348 . . . . . . 7 ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎) → ((𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) ↔ (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
1918biimprcd 249 . . . . . 6 ((∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎) → (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
2016, 17, 19e10 43455 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
21 pm2.27 42 . . . . 5 (𝑎 ≠ ∅ → ((𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
223, 20, 21e11 43449 . . . 4 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
2322in1 43332 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
2423ax-gen 1798 . 2 𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
25 dfepfr 5662 . . 3 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2625biimpri 227 . 2 (∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅) → E Fr On)
2724, 26e0a 43533 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  wal 1540   = wceq 1542  wex 1782  wcel 2107  wne 2941  wrex 3071  cin 3948  wss 3949  c0 4323   E cep 5580   Fr wfr 5629  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-13 2372  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  df-vd3 43351
This theorem is referenced by: (None)
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