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Theorem onfr 5925
Description: The ordinal class is well-founded. This lemma is needed for ordon 7149 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 5252 . 2 ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅))
2 n0 4075 . . . 4 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
3 ineq2 3952 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑥𝑧) = (𝑥𝑦))
43eqeq1d 2763 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑥𝑧) = ∅ ↔ (𝑥𝑦) = ∅))
54rspcev 3450 . . . . . . . 8 ((𝑦𝑥 ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
65adantll 752 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) = ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
7 inss1 3977 . . . . . . . 8 (𝑥𝑦) ⊆ 𝑥
8 ssel2 3740 . . . . . . . . . . 11 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
9 eloni 5895 . . . . . . . . . . 11 (𝑦 ∈ On → Ord 𝑦)
10 ordfr 5900 . . . . . . . . . . 11 (Ord 𝑦 → E Fr 𝑦)
118, 9, 103syl 18 . . . . . . . . . 10 ((𝑥 ⊆ On ∧ 𝑦𝑥) → E Fr 𝑦)
12 inss2 3978 . . . . . . . . . . 11 (𝑥𝑦) ⊆ 𝑦
13 vex 3344 . . . . . . . . . . . . 13 𝑥 ∈ V
1413inex1 4952 . . . . . . . . . . . 12 (𝑥𝑦) ∈ V
1514epfrc 5253 . . . . . . . . . . 11 (( E Fr 𝑦 ∧ (𝑥𝑦) ⊆ 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1612, 15mp3an2 1561 . . . . . . . . . 10 (( E Fr 𝑦 ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
1711, 16sylan 489 . . . . . . . . 9 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅)
18 inass 3967 . . . . . . . . . . . . 13 ((𝑥𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦𝑧))
198, 9syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ⊆ On ∧ 𝑦𝑥) → Ord 𝑦)
20 elinel2 3944 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑥𝑦) → 𝑧𝑦)
21 ordelss 5901 . . . . . . . . . . . . . . . 16 ((Ord 𝑦𝑧𝑦) → 𝑧𝑦)
2219, 20, 21syl2an 495 . . . . . . . . . . . . . . 15 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧𝑦)
23 sseqin2 3961 . . . . . . . . . . . . . . 15 (𝑧𝑦 ↔ (𝑦𝑧) = 𝑧)
2422, 23sylib 208 . . . . . . . . . . . . . 14 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑦𝑧) = 𝑧)
2524ineq2d 3958 . . . . . . . . . . . . 13 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑥 ∩ (𝑦𝑧)) = (𝑥𝑧))
2618, 25syl5eq 2807 . . . . . . . . . . . 12 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → ((𝑥𝑦) ∩ 𝑧) = (𝑥𝑧))
2726eqeq1d 2763 . . . . . . . . . . 11 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ 𝑧 ∈ (𝑥𝑦)) → (((𝑥𝑦) ∩ 𝑧) = ∅ ↔ (𝑥𝑧) = ∅))
2827rexbidva 3188 . . . . . . . . . 10 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅))
2928adantr 472 . . . . . . . . 9 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → (∃𝑧 ∈ (𝑥𝑦)((𝑥𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅))
3017, 29mpbid 222 . . . . . . . 8 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅)
31 ssrexv 3809 . . . . . . . 8 ((𝑥𝑦) ⊆ 𝑥 → (∃𝑧 ∈ (𝑥𝑦)(𝑥𝑧) = ∅ → ∃𝑧𝑥 (𝑥𝑧) = ∅))
327, 30, 31mpsyl 68 . . . . . . 7 (((𝑥 ⊆ On ∧ 𝑦𝑥) ∧ (𝑥𝑦) ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
336, 32pm2.61dane 3020 . . . . . 6 ((𝑥 ⊆ On ∧ 𝑦𝑥) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
3433ex 449 . . . . 5 (𝑥 ⊆ On → (𝑦𝑥 → ∃𝑧𝑥 (𝑥𝑧) = ∅))
3534exlimdv 2011 . . . 4 (𝑥 ⊆ On → (∃𝑦 𝑦𝑥 → ∃𝑧𝑥 (𝑥𝑧) = ∅))
362, 35syl5bi 232 . . 3 (𝑥 ⊆ On → (𝑥 ≠ ∅ → ∃𝑧𝑥 (𝑥𝑧) = ∅))
3736imp 444 . 2 ((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧𝑥 (𝑥𝑧) = ∅)
381, 37mpgbir 1875 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2140  wne 2933  wrex 3052  cin 3715  wss 3716  c0 4059   E cep 5179   Fr wfr 5223  Ord word 5884  Oncon0 5885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-tr 4906  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-ord 5888  df-on 5889
This theorem is referenced by:  ordon  7149
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