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Theorem onfr 6423
Description: The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7797 (through epweon 7795) 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 5669 . 2 ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅))
2 n0 4353 . . . 4 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
3 ineq2 4214 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑥𝑧) = (𝑥𝑦))
43eqeq1d 2739 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝑧) = ∅ ↔ (𝑥𝑦) = ∅))
54rspcev 3622 . . . . . . . 8 ((𝑦𝑥 ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
65adantll 714 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
7 inss1 4237 . . . . . . . 8 (𝑥𝑦) ⊆ 𝑥
8 ssel2 3978 . . . . . . . . . . 11 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
9 eloni 6394 . . . . . . . . . . 11 (𝑦 ∈ On → Ord 𝑦)
10 ordfr 6399 . . . . . . . . . . 11 (Ord 𝑦 → E Fr 𝑦)
118, 9, 103syl 18 . . . . . . . . . 10 ((𝑥 ⊆ On ∧ 𝑦𝑥) → E Fr 𝑦)
12 inss2 4238 . . . . . . . . . . 11 (𝑥𝑦) ⊆ 𝑦
13 vex 3484 . . . . . . . . . . . . 13 𝑥 ∈ V
1413inex1 5317 . . . . . . . . . . . 12 (𝑥𝑦) ∈ V
1514epfrc 5670 . . . . . . . . . . 11 (( E Fr 𝑦 ∧ (𝑥𝑦) ⊆ 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1612, 15mp3an2 1451 . . . . . . . . . 10 (( E Fr 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1711, 16sylan 580 . . . . . . . . 9 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
18 inass 4228 . . . . . . . . . . . . 13 ((𝑥𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦𝑧))
198, 9syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ⊆ On ∧ 𝑦𝑥) → Ord 𝑦)
20 elinel2 4202 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑥𝑦) → 𝑧𝑦)
21 ordelss 6400 . . . . . . . . . . . . . . . 16 ((Ord 𝑦𝑧𝑦) → 𝑧𝑦)
2219, 20, 21syl2an 596 . . . . . . . . . . . . . . 15 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧𝑦)
23 sseqin2 4223 . . . . . . . . . . . . . . 15 (𝑧𝑦 ↔ (𝑦𝑧) = 𝑧)
2422, 23sylib 218 . . . . . . . . . . . . . 14 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑦𝑧) = 𝑧)
2524ineq2d 4220 . . . . . . . . . . . . 13 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑥 ∩ (𝑦𝑧)) = (𝑥𝑧))
2618, 25eqtrid 2789 . . . . . . . . . . . 12 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → ((𝑥𝑦) ∩ 𝑧) = (𝑥𝑧))
2726eqeq1d 2739 . . . . . . . . . . 11 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (((𝑥𝑦) ∩ 𝑧) = ∅ ↔ (𝑥𝑧) = ∅))
2827rexbidva 3177 . . . . . . . . . 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 4053 . . . . . . . 8 ((𝑥𝑦) ⊆ 𝑥 → (∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅ → ∃𝑧𝑥 (𝑥𝑧) = ∅))
327, 30, 31mpsyl 68 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
336, 32pm2.61dane 3029 . . . . . 6 ((𝑥 ⊆ On ∧ 𝑦𝑥) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
3433ex 412 . . . . 5 (𝑥 ⊆ On → (𝑦𝑥 → ∃𝑧𝑥 (𝑥𝑧) = ∅))
3534exlimdv 1933 . . . 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 1799 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  cin 3950  wss 3951  c0 4333   E cep 5583   Fr wfr 5634  Ord word 6383  Oncon0 6384
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  epweon  7795  epweonALT  7796  on2recsfn  8705  on2recsov  8706  on2ind  8707  on3ind  8708  wffr  44978
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