Theorem epweon 7708

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 7668, see epweonALT 7709. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7668. (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 6345	. 2 ⊢ E Fr On
2		df-po 5524	. . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
3		eloni 6316	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordirr 6324	. . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥)
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥)
6		epel 5519	. . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
7	5, 6	sylnibr 329	. . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥)
8		ontr1 6353	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 5519	. . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 5519	. . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 628	. . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 5519	. . . . . . . 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 3124	. . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
16	15	ralrimivw 3128	. . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
17	2, 16	mprgbir 3054	. . 3 ⊢ E Po On
18		eloni 6316	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
19		ordtri3or 6338	. . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
20		biid 261	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
21		epel 5519	. . . . . . 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 3172	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
26		df-so 5525	. . 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 5571	. 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 2111  ∀wral 3047   class class class wbr 5091   E cep 5515   Po wpo 5522   Or wor 5523   Fr wfr 5566   We wwe 5568  Ord word 6305  Oncon0 6306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310

This theorem is referenced by:  ordon  7710  dford5  7717  omsinds  7817  onnseq  8264  dfrecs3  8292  tfr1ALT  8319  tfr2ALT  8320  tfr3ALT  8321  on2recsfn  8582  on2recsov  8583  on2ind  8584  on3ind  8585  ordunifi  9174  ordtypelem8  9411  oismo  9426  cantnfcl  9557  leweon  9899  r0weon  9900  ac10ct  9922  dfac12lem2  10033  cflim2  10151  cofsmo  10157  hsmexlem1  10314  smobeth  10474  gruina  10706  ltsopi  10776  onswe  28204  finminlem  36351  dnwech  43080  aomclem4  43089  onsupuni  43261  oninfint  43268  epsoon  43285  epirron  43286  oneptr  43287  oaun3lem1  43406