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Theorem onfr 6415
Description: The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7785 (through epweon 7783) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
onfr E Fr On

Proof of Theorem onfr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfepfr 5667 . 2 ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅))
2 n0 4349 . . . 4 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
3 ineq2 4207 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑥𝑧) = (𝑥𝑦))
43eqeq1d 2728 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝑧) = ∅ ↔ (𝑥𝑦) = ∅))
54rspcev 3608 . . . . . . . 8 ((𝑦𝑥 ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
65adantll 712 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
7 inss1 4230 . . . . . . . 8 (𝑥𝑦) ⊆ 𝑥
8 ssel2 3974 . . . . . . . . . . 11 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
9 eloni 6386 . . . . . . . . . . 11 (𝑦 ∈ On → Ord 𝑦)
10 ordfr 6391 . . . . . . . . . . 11 (Ord 𝑦 → E Fr 𝑦)
118, 9, 103syl 18 . . . . . . . . . 10 ((𝑥 ⊆ On ∧ 𝑦𝑥) → E Fr 𝑦)
12 inss2 4231 . . . . . . . . . . 11 (𝑥𝑦) ⊆ 𝑦
13 vex 3466 . . . . . . . . . . . . 13 𝑥 ∈ V
1413inex1 5322 . . . . . . . . . . . 12 (𝑥𝑦) ∈ V
1514epfrc 5668 . . . . . . . . . . 11 (( E Fr 𝑦 ∧ (𝑥𝑦) ⊆ 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1612, 15mp3an2 1446 . . . . . . . . . 10 (( E Fr 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1711, 16sylan 578 . . . . . . . . 9 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
18 inass 4221 . . . . . . . . . . . . 13 ((𝑥𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦𝑧))
198, 9syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ⊆ On ∧ 𝑦𝑥) → Ord 𝑦)
20 elinel2 4197 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑥𝑦) → 𝑧𝑦)
21 ordelss 6392 . . . . . . . . . . . . . . . 16 ((Ord 𝑦𝑧𝑦) → 𝑧𝑦)
2219, 20, 21syl2an 594 . . . . . . . . . . . . . . 15 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧𝑦)
23 sseqin2 4216 . . . . . . . . . . . . . . 15 (𝑧𝑦 ↔ (𝑦𝑧) = 𝑧)
2422, 23sylib 217 . . . . . . . . . . . . . 14 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑦𝑧) = 𝑧)
2524ineq2d 4213 . . . . . . . . . . . . 13 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑥 ∩ (𝑦𝑧)) = (𝑥𝑧))
2618, 25eqtrid 2778 . . . . . . . . . . . 12 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → ((𝑥𝑦) ∩ 𝑧) = (𝑥𝑧))
2726eqeq1d 2728 . . . . . . . . . . 11 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (((𝑥𝑦) ∩ 𝑧) = ∅ ↔ (𝑥𝑧) = ∅))
2827rexbidva 3167 . . . . . . . . . 10 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅))
2928adantr 479 . . . . . . . . 9 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → (∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅))
3017, 29mpbid 231 . . . . . . . 8 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅)
31 ssrexv 4049 . . . . . . . 8 ((𝑥𝑦) ⊆ 𝑥 → (∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅ → ∃𝑧𝑥 (𝑥𝑧) = ∅))
327, 30, 31mpsyl 68 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
336, 32pm2.61dane 3019 . . . . . 6 ((𝑥 ⊆ On ∧ 𝑦𝑥) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
3433ex 411 . . . . 5 (𝑥 ⊆ On → (𝑦𝑥 → ∃𝑧𝑥 (𝑥𝑧) = ∅))
3534exlimdv 1929 . . . 4 (𝑥 ⊆ On → (∃𝑦 𝑦𝑥 → ∃𝑧𝑥 (𝑥𝑧) = ∅))
362, 35biimtrid 241 . . 3 (𝑥 ⊆ On → (𝑥 ≠ ∅ → ∃𝑧𝑥 (𝑥𝑧) = ∅))
3736imp 405 . 2 ((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
381, 37mpgbir 1794 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  wne 2930  wrex 3060  cin 3946  wss 3947  c0 4325   E cep 5585   Fr wfr 5634  Ord word 6375  Oncon0 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380
This theorem is referenced by:  epweon  7783  epweonALT  7784  on2recsfn  8697  on2recsov  8698  on2ind  8699  on3ind  8700
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