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Theorem onfr 6425
Description: The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7796 (through epweon 7794) 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 5673 . 2 ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅))
2 n0 4359 . . . 4 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
3 ineq2 4222 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑥𝑧) = (𝑥𝑦))
43eqeq1d 2737 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝑧) = ∅ ↔ (𝑥𝑦) = ∅))
54rspcev 3622 . . . . . . . 8 ((𝑦𝑥 ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
65adantll 714 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
7 inss1 4245 . . . . . . . 8 (𝑥𝑦) ⊆ 𝑥
8 ssel2 3990 . . . . . . . . . . 11 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
9 eloni 6396 . . . . . . . . . . 11 (𝑦 ∈ On → Ord 𝑦)
10 ordfr 6401 . . . . . . . . . . 11 (Ord 𝑦 → E Fr 𝑦)
118, 9, 103syl 18 . . . . . . . . . 10 ((𝑥 ⊆ On ∧ 𝑦𝑥) → E Fr 𝑦)
12 inss2 4246 . . . . . . . . . . 11 (𝑥𝑦) ⊆ 𝑦
13 vex 3482 . . . . . . . . . . . . 13 𝑥 ∈ V
1413inex1 5323 . . . . . . . . . . . 12 (𝑥𝑦) ∈ V
1514epfrc 5674 . . . . . . . . . . 11 (( E Fr 𝑦 ∧ (𝑥𝑦) ⊆ 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1612, 15mp3an2 1448 . . . . . . . . . 10 (( E Fr 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1711, 16sylan 580 . . . . . . . . 9 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
18 inass 4236 . . . . . . . . . . . . 13 ((𝑥𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦𝑧))
198, 9syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ⊆ On ∧ 𝑦𝑥) → Ord 𝑦)
20 elinel2 4212 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑥𝑦) → 𝑧𝑦)
21 ordelss 6402 . . . . . . . . . . . . . . . 16 ((Ord 𝑦𝑧𝑦) → 𝑧𝑦)
2219, 20, 21syl2an 596 . . . . . . . . . . . . . . 15 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧𝑦)
23 sseqin2 4231 . . . . . . . . . . . . . . 15 (𝑧𝑦 ↔ (𝑦𝑧) = 𝑧)
2422, 23sylib 218 . . . . . . . . . . . . . 14 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑦𝑧) = 𝑧)
2524ineq2d 4228 . . . . . . . . . . . . 13 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑥 ∩ (𝑦𝑧)) = (𝑥𝑧))
2618, 25eqtrid 2787 . . . . . . . . . . . 12 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → ((𝑥𝑦) ∩ 𝑧) = (𝑥𝑧))
2726eqeq1d 2737 . . . . . . . . . . 11 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (((𝑥𝑦) ∩ 𝑧) = ∅ ↔ (𝑥𝑧) = ∅))
2827rexbidva 3175 . . . . . . . . . 10 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅))
2928adantr 480 . . . . . . . . 9 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → (∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅))
3017, 29mpbid 232 . . . . . . . 8 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅)
31 ssrexv 4065 . . . . . . . 8 ((𝑥𝑦) ⊆ 𝑥 → (∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅ → ∃𝑧𝑥 (𝑥𝑧) = ∅))
327, 30, 31mpsyl 68 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
336, 32pm2.61dane 3027 . . . . . 6 ((𝑥 ⊆ On ∧ 𝑦𝑥) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
3433ex 412 . . . . 5 (𝑥 ⊆ On → (𝑦𝑥 → ∃𝑧𝑥 (𝑥𝑧) = ∅))
3534exlimdv 1931 . . . 4 (𝑥 ⊆ On → (∃𝑦 𝑦𝑥 → ∃𝑧𝑥 (𝑥𝑧) = ∅))
362, 35biimtrid 242 . . 3 (𝑥 ⊆ On → (𝑥 ≠ ∅ → ∃𝑧𝑥 (𝑥𝑧) = ∅))
3736imp 406 . 2 ((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
381, 37mpgbir 1796 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wrex 3068  cin 3962  wss 3963  c0 4339   E cep 5588   Fr wfr 5638  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  epweon  7794  epweonALT  7795  on2recsfn  8704  on2recsov  8705  on2ind  8706  on3ind  8707
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