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Theorem onfrALTVD 41532
Description: Virtual deduction proof of onfrALT 41190. 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 41190 is onfrALTVD 41532 without virtual deductions and was automatically derived from onfrALTVD 41532.
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 41215 . . . . . 6 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2 simpr 488 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
31, 2e1a 41268 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
4 exmid 892 . . . . . . . . . 10 ((𝑎𝑥) = ∅ ∨ ¬ (𝑎𝑥) = ∅)
5 onfrALTlem1VD 41531 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
65in2an 41249 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
7 onfrALTlem2VD 41530 . . . . . . . . . . 11 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
87in2an 41249 . . . . . . . . . 10 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
9 pm2.61 195 . . . . . . . . . . 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 41282 . . . . . . . . 9 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥𝑎   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
1211in2 41246 . . . . . . . 8 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
1312gen11 41257 . . . . . . 7 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
14 19.23v 1943 . . . . . . . 8 (∀𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) ↔ (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
1514biimpi 219 . . . . . . 7 (∀𝑥(𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) → (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
1613, 15e1a 41268 . . . . . 6 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
17 n0 4282 . . . . . 6 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
18 imbi1 351 . . . . . . 7 ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎) → ((𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) ↔ (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
1918biimprcd 253 . . . . . 6 ((∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎) → (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
2016, 17, 19e10 41335 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)   )
21 pm2.27 42 . . . . 5 (𝑎 ≠ ∅ → ((𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
223, 20, 21e11 41329 . . . 4 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
2322in1 41212 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
2423ax-gen 1797 . 2 𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
25 dfepfr 5517 . . 3 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2625biimpri 231 . 2 (∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅) → E Fr On)
2724, 26e0a 41413 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  wal 1536   = wceq 1538  wex 1781  wcel 2114  wne 3011  wrex 3131  cin 3907  wss 3908  c0 4265   E cep 5441   Fr wfr 5488  Oncon0 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-vd1 41211  df-vd2 41219  df-vd3 41231
This theorem is referenced by: (None)
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