Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onfrALTlem2 Structured version   Visualization version   GIF version

Theorem onfrALTlem2 44543
Description: Lemma for onfrALT 44546. (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 484 . . . . . . . . . . . 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 4245 . . . . . . . . . . . 12 (𝑎𝑦) ⊆ 𝑦
43sseli 3990 . . . . . . . . . . 11 (𝑧 ∈ (𝑎𝑦) → 𝑧𝑦)
52, 4syl8 76 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧𝑦)))
6 inss1 4244 . . . . . . . . . . . . 13 (𝑎𝑦) ⊆ 𝑎
76sseli 3990 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎𝑦) → 𝑧𝑎)
82, 7syl8 76 . . . . . . . . . . 11 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧𝑎)))
9 simpl 482 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
10 simpl 482 . . . . . . . . . . . . . . 15 ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥𝑎)
11 ssel 3988 . . . . . . . . . . . . . . 15 (𝑎 ⊆ On → (𝑥𝑎𝑥 ∈ On))
129, 10, 11syl2im 40 . . . . . . . . . . . . . 14 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → 𝑥 ∈ On))
13 eloni 6395 . . . . . . . . . . . . . 14 (𝑥 ∈ On → Ord 𝑥)
1412, 13syl6 35 . . . . . . . . . . . . 13 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Ord 𝑥))
15 ordtr 6399 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
1614, 15syl6 35 . . . . . . . . . . . 12 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → Tr 𝑥))
17 simpll 767 . . . . . . . . . . . . . 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 4245 . . . . . . . . . . . . . 14 (𝑎𝑥) ⊆ 𝑥
2019sseli 3990 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑎𝑥) → 𝑦𝑥)
2118, 20syl8 76 . . . . . . . . . . . 12 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑦𝑥)))
22 trel 5273 . . . . . . . . . . . . 13 (Tr 𝑥 → ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2322expcomd 416 . . . . . . . . . . . 12 (Tr 𝑥 → (𝑦𝑥 → (𝑧𝑦𝑧𝑥)))
2416, 21, 5, 23ee233 44516 . . . . . . . . . . 11 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧𝑥)))
25 elin 3978 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎𝑥) ↔ (𝑧𝑎𝑧𝑥))
2625simplbi2 500 . . . . . . . . . . 11 (𝑧𝑎 → (𝑧𝑥𝑧 ∈ (𝑎𝑥)))
278, 24, 26ee33 44518 . . . . . . . . . 10 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧 ∈ (𝑎𝑥))))
28 elin 3978 . . . . . . . . . . 11 (𝑧 ∈ ((𝑎𝑥) ∩ 𝑦) ↔ (𝑧 ∈ (𝑎𝑥) ∧ 𝑧𝑦))
2928simplbi2com 502 . . . . . . . . . 10 (𝑧𝑦 → (𝑧 ∈ (𝑎𝑥) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))
305, 27, 29ee33 44518 . . . . . . . . 9 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → (((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎𝑦)) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦))))
3130exp4a 431 . . . . . . . 8 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))))
3231ggen31 44542 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → ∀𝑧(𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))))
33 df-ss 3979 . . . . . . . 8 ((𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦) ↔ ∀𝑧(𝑧 ∈ (𝑎𝑦) → 𝑧 ∈ ((𝑎𝑥) ∩ 𝑦)))
3433biimpri 228 . . . . . . 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 484 . . . . . . 7 ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → ((𝑎𝑥) ∩ 𝑦) = ∅)
37362a1i 12 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → ((𝑎𝑥) ∩ 𝑦) = ∅)))
38 sseq0 4408 . . . . . . 7 (((𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑎𝑦) = ∅)
3938ex 412 . . . . . 6 ((𝑎𝑦) ⊆ ((𝑎𝑥) ∩ 𝑦) → (((𝑎𝑥) ∩ 𝑦) = ∅ → (𝑎𝑦) = ∅))
4035, 37, 39ee33 44518 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑎𝑦) = ∅)))
41 simpl 482 . . . . . . 7 ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎𝑥))
42412a1i 12 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎𝑥))))
43 inss1 4244 . . . . . . 7 (𝑎𝑥) ⊆ 𝑎
4443sseli 3990 . . . . . 6 (𝑦 ∈ (𝑎𝑥) → 𝑦𝑎)
4542, 44syl8 76 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → 𝑦𝑎)))
46 pm3.21 471 . . . . 5 ((𝑎𝑦) = ∅ → (𝑦𝑎 → (𝑦𝑎 ∧ (𝑎𝑦) = ∅)))
4740, 45, 46ee33 44518 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑦𝑎 ∧ (𝑎𝑦) = ∅))))
4847alrimdv 1926 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∀𝑦((𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅) → (𝑦𝑎 ∧ (𝑎𝑦) = ∅))))
49 onfrALTlem3 44541 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))
50 df-rex 3068 . . . 4 (∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅))
5149, 50imbitrdi 251 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦(𝑦 ∈ (𝑎𝑥) ∧ ((𝑎𝑥) ∩ 𝑦) = ∅)))
52 exim 1830 . . 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 3068 . 2 (∃𝑦𝑎 (𝑎𝑦) = ∅ ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
5553, 54imbitrrdi 252 1 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1534   = wceq 1536  wex 1775  wcel 2105  wne 2937  wrex 3067  cin 3961  wss 3962  c0 4338  Tr wtr 5264  Ord word 6384  Oncon0 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389
This theorem is referenced by:  onfrALT  44546
  Copyright terms: Public domain W3C validator