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Theorem onfrALTVD 39643
Description: Virtual deduction proof of onfrALT 39283. 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 39283 is onfrALTVD 39643 without virtual deductions and was automatically derived from onfrALTVD 39643.
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 39309 . . . . . 6 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2 simpr 471 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
31, 2e1a 39371 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
4 exmid 860 . . . . . . . . . 10 ((𝑎𝑥) = ∅ ∨ ¬ (𝑎𝑥) = ∅)
5 onfrALTlem1VD 39642 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
65in2an 39352 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
7 onfrALTlem2VD 39641 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
87in2an 39352 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
9 pm2.61 183 . . . . . . . . . . 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 39385 . . . . . . . . 9 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
1211in2 39349 . . . . . . . 8 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
1312gen11 39360 . . . . . . 7 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
14 19.23v 2022 . . . . . . . 8 (∀𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) ↔ (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
1514biimpi 206 . . . . . . 7 (∀𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) → (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
1613, 15e1a 39371 . . . . . 6 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
17 n0 4076 . . . . . 6 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
18 imbi1 336 . . . . . . 7 ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎) → ((𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) ↔ (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
1918biimprcd 240 . . . . . 6 ((∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎) → (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
2016, 17, 19e10 39438 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
21 pm2.27 42 . . . . 5 (𝑎 ≠ ∅ → ((𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
223, 20, 21e11 39432 . . . 4 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
2322in1 39306 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
2423ax-gen 1869 . 2 𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
25 dfepfr 5234 . . 3 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2625biimpri 218 . 2 (∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅) → E Fr On)
2724, 26e0a 39518 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  wal 1628   = wceq 1630  wex 1851  wcel 2144  wne 2942  wrex 3061  cin 3720  wss 3721  c0 4061   E cep 5161   Fr wfr 5205  Oncon0 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-tr 4885  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-vd1 39305  df-vd2 39313  df-vd3 39325
This theorem is referenced by: (None)
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