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Theorem onfrALTlem2 40886
Description: Lemma for onfrALT 40889. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
Distinct variable groups:   𝑦,𝑎   𝑥,𝑦

Proof of Theorem onfrALTlem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simpr 487 . . . . . . . . . . . 12 (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧 ∈ (𝑎𝑦))
212a1i 12 . . . . . . . . . . 11 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧 ∈ (𝑎𝑦))))
3 inss2 4209 . . . . . . . . . . . 12 (𝑎𝑦) ⊆ 𝑦
43sseli 3966 . . . . . . . . . . 11 (𝑧 ∈ (𝑎𝑦) → 𝑧𝑦)
52, 4syl8 76 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧𝑦)))
6 inss1 4208 . . . . . . . . . . . . 13 (𝑎𝑦) ⊆ 𝑎
76sseli 3966 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎𝑦) → 𝑧𝑎)
82, 7syl8 76 . . . . . . . . . . 11 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧𝑎)))
9 simpl 485 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
10 simpl 485 . . . . . . . . . . . . . . 15 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥𝑎)
11 ssel 3964 . . . . . . . . . . . . . . 15 (𝑎 ⊆ On → (𝑥𝑎𝑥 ∈ On))
129, 10, 11syl2im 40 . . . . . . . . . . . . . 14 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥 ∈ On))
13 eloni 6204 . . . . . . . . . . . . . 14 (𝑥 ∈ On → Ord 𝑥)
1412, 13syl6 35 . . . . . . . . . . . . 13 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Ord 𝑥))
15 ordtr 6208 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
1614, 15syl6 35 . . . . . . . . . . . 12 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Tr 𝑥))
17 simpll 765 . . . . . . . . . . . . . 14 (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑦 ∈ (𝑎𝑥))
18172a1i 12 . . . . . . . . . . . . 13 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑦 ∈ (𝑎𝑥))))
19 inss2 4209 . . . . . . . . . . . . . 14 (𝑎𝑥) ⊆ 𝑥
2019sseli 3966 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑎𝑥) → 𝑦𝑥)
2118, 20syl8 76 . . . . . . . . . . . 12 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑦𝑥)))
22 trel 5182 . . . . . . . . . . . . 13 (Tr 𝑥 → ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2322expcomd 419 . . . . . . . . . . . 12 (Tr 𝑥 → (𝑦𝑥 → (𝑧𝑦𝑧𝑥)))
2416, 21, 5, 23ee233 40859 . . . . . . . . . . 11 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧𝑥)))
25 elin 4172 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎𝑥) ↔ (𝑧𝑎𝑧𝑥))
2625simplbi2 503 . . . . . . . . . . 11 (𝑧𝑎 → (𝑧𝑥𝑧 ∈ (𝑎𝑥)))
278, 24, 26ee33 40861 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧 ∈ (𝑎𝑥))))
28 elin 4172 . . . . . . . . . . 11 (𝑧 ∈ ((𝑎𝑥) ∩ 𝑦) ↔ (𝑧 ∈ (𝑎𝑥) ∧ 𝑧𝑦))
2928simplbi2com 505 . . . . . . . . . 10 (𝑧𝑦 → (𝑧 ∈ (𝑎𝑥) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))
305, 27, 29ee33 40861 . . . . . . . . 9 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦))))
3130exp4a 434 . . . . . . . 8 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))))
3231ggen31 40885 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → ∀𝑧(𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))))
33 dfss2 3958 . . . . . . . 8 ((𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦) ↔ ∀𝑧(𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))
3433biimpri 230 . . . . . . 7 (∀𝑧(𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)) → (𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦))
3532, 34syl8 76 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦))))
36 simpr 487 . . . . . . 7 ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → ((𝑎𝑥) ∩ 𝑦) = ∅)
37362a1i 12 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → ((𝑎𝑥) ∩ 𝑦) = ∅)))
38 sseq0 4356 . . . . . . 7 (((𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑎𝑦) = ∅)
3938ex 415 . . . . . 6 ((𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦) → (((𝑎𝑥) ∩ 𝑦) = ∅ → (𝑎𝑦) = ∅))
4035, 37, 39ee33 40861 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑎𝑦) = ∅)))
41 simpl 485 . . . . . . 7 ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎𝑥))
42412a1i 12 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎𝑥))))
43 inss1 4208 . . . . . . 7 (𝑎𝑥) ⊆ 𝑎
4443sseli 3966 . . . . . 6 (𝑦 ∈ (𝑎𝑥) → 𝑦𝑎)
4542, 44syl8 76 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → 𝑦𝑎)))
46 pm3.21 474 . . . . 5 ((𝑎𝑦) = ∅ → (𝑦𝑎 → (𝑦𝑎 ∧ (𝑎𝑦) = ∅)))
4740, 45, 46ee33 40861 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑦𝑎 ∧ (𝑎𝑦) = ∅))))
4847alrimdv 1929 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∀𝑦((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑦𝑎 ∧ (𝑎𝑦) = ∅))))
49 onfrALTlem3 40884 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
50 df-rex 3147 . . . 4 (∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅))
5149, 50syl6ib 253 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅)))
52 exim 1833 . . 3 (∀𝑦((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑦𝑎 ∧ (𝑎𝑦) = ∅)) → (∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)))
5348, 51, 52syl6c 70 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)))
54 df-rex 3147 . 2 (∃𝑦𝑎 (𝑎𝑦) = ∅ ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
5553, 54syl6ibr 254 1 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1534   = wceq 1536  wex 1779  wcel 2113  wne 3019  wrex 3142  cin 3938  wss 3939  c0 4294  Tr wtr 5175  Ord word 6193  Oncon0 6194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-tr 5176  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-ord 6197  df-on 6198
This theorem is referenced by:  onfrALT  40889
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