Theorem epweon 7751

Description: The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. For a shorter proof requiring ax-un 7711, see epweonALT 7752. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7711. (Revised by BTernaryTau, 30-Nov-2024.)

Assertion

Ref	Expression
epweon	⊢ E We On

Proof of Theorem epweon

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onfr 6371	. 2 ⊢ E Fr On
2		df-po 5546	. . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
3		eloni 6342	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordirr 6350	. . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥)
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥)
6		epel 5541	. . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
7	5, 6	sylnibr 329	. . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥)
8		ontr1 6379	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 5541	. . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 5541	. . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 628	. . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 5541	. . . . . . . 8 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
13	8, 11, 12	3imtr4g 296	. . . . . . 7 ⊢ (𝑧 ∈ On → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
14	7, 13	anim12i 613	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
15	14	ralrimiva 3125	. . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
16	15	ralrimivw 3129	. . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
17	2, 16	mprgbir 3051	. . 3 ⊢ E Po On
18		eloni 6342	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
19		ordtri3or 6364	. . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
20		biid 261	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
21		epel 5541	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
22	9, 20, 21	3orbi123i 1156	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
23	19, 22	sylibr 234	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
24	3, 18, 23	syl2an 596	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
25	24	rgen2 3177	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
26		df-so 5547	. . 3 ⊢ ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
27	17, 25, 26	mpbir2an 711	. 2 ⊢ E Or On
28		df-we 5593	. 2 ⊢ ( E We On ↔ ( E Fr On ∧ E Or On))
29	1, 27, 28	mpbir2an 711	1 ⊢ E We On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ w3o 1085   ∈ wcel 2109  ∀wral 3044   class class class wbr 5107   E cep 5537   Po wpo 5544   Or wor 5545   Fr wfr 5588   We wwe 5590  Ord word 6331  Oncon0 6332

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336

This theorem is referenced by:  ordon  7753  dford5  7760  omsinds  7863  onnseq  8313  dfrecs3  8341  tfr1ALT  8368  tfr2ALT  8369  tfr3ALT  8370  on2recsfn  8631  on2recsov  8632  on2ind  8633  on3ind  8634  ordunifi  9237  ordtypelem8  9478  oismo  9493  cantnfcl  9620  leweon  9964  r0weon  9965  ac10ct  9987  dfac12lem2  10098  cflim2  10216  cofsmo  10222  hsmexlem1  10379  smobeth  10539  gruina  10771  ltsopi  10841  onswe  28170  finminlem  36306  dnwech  43037  aomclem4  43046  onsupuni  43218  oninfint  43225  epsoon  43242  epirron  43243  oneptr  43244  oaun3lem1  43363