Theorem epweon 7778

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 7740, see epweonALT 7779. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7740. (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 6401	. 2 ⊢ E Fr On
2		df-po 5567	. . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
3		eloni 6371	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordirr 6379	. . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥)
5	3, 4	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥)
6		epel 5562	. . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
7	5, 6	sylnibr 332	. . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥)
8		ontr1 6409	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 5562	. . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 5562	. . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 640	. . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 5562	. . . . . . . 8 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
13	8, 11, 12	3imtr4g 299	. . . . . . 7 ⊢ (𝑧 ∈ On → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
14	7, 13	anim12i 625	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
15	14	ralrimiva 3156	. . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
16	15	ralrimivw 3160	. . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
17	2, 16	mprgbir 3085	. . 3 ⊢ E Po On
18		eloni 6371	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
19		ordtri3or 6394	. . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
20		biid 264	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
21		epel 5562	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
22	9, 20, 21	3orbi123i 1174	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
23	19, 22	sylibr 237	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
24	3, 18, 23	syl2an 608	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
25	24	rgen2 3204	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
26		df-so 5568	. . 3 ⊢ ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
27	17, 25, 26	mpbir2an 724	. 2 ⊢ E Or On
28		df-we 5614	. 2 ⊢ ( E We On ↔ ( E Fr On ∧ E Or On))
29	1, 27, 28	mpbir2an 724	1 ⊢ E We On

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ w3o 1102   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107   E cep 5558   Po wpo 5565   Or wor 5566   Fr wfr 5609   We wwe 5611  Ord word 6360  Oncon0 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365

This theorem is used by:  ordon  7780  dford5  7787  omsinds  7887  onnseq  8337  dfrecs3  8365  tfr1ALT  8393  tfr2ALT  8394  tfr3ALT  8395  on2recsfn  8659  on2recsov  8660  on2ind  8661  on3ind  8662  ordunifi  9264  ordtypelem8  9501  oismo  9516  cantnfcl  9650  leweon  10018  r0weon  10019  ac10ct  10041  dfac12lem2  10151  cflim2  10269  cofsmo  10275  hsmexlem1  10432  smobeth  10599  gruina  10831  ltsopi  10901  onswe  28545  finminlem  36945  dnwech  43897  aomclem4  43906  onsupuni  44078  oninfint  44085  epsoon  44102  epirron  44103  oneptr  44104  oaun3lem1  44223