Theorem epweon 7749

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 7712, see epweonALT 7750. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7712. (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 6395	. 2 ⊢ E Fr On
2		df-po 5584	. . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
3		eloni 6366	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordirr 6374	. . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥)
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥)
6		epel 5579	. . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
7	5, 6	sylnibr 329	. . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥)
8		ontr1 6402	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 5579	. . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 5579	. . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 628	. . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 5579	. . . . . . . 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 614	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
15	14	ralrimiva 3147	. . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
16	15	ralrimivw 3151	. . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
17	2, 16	mprgbir 3069	. . 3 ⊢ E Po On
18		eloni 6366	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
19		ordtri3or 6388	. . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
20		biid 261	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
21		epel 5579	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
22	9, 20, 21	3orbi123i 1157	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
23	19, 22	sylibr 233	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
24	3, 18, 23	syl2an 597	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
25	24	rgen2 3198	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
26		df-so 5585	. . 3 ⊢ ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
27	17, 25, 26	mpbir2an 710	. 2 ⊢ E Or On
28		df-we 5629	. 2 ⊢ ( E We On ↔ ( E Fr On ∧ E Or On))
29	1, 27, 28	mpbir2an 710	1 ⊢ E We On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ w3o 1087   ∈ wcel 2107  ∀wral 3062   class class class wbr 5144   E cep 5575   Po wpo 5582   Or wor 5583   Fr wfr 5624   We wwe 5626  Ord word 6355  Oncon0 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-tr 5262  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6359  df-on 6360

This theorem is referenced by:  ordon  7751  dford5  7758  omsinds  7863  omsindsOLD  7864  onnseq  8331  dfrecs3  8359  dfrecs3OLD  8360  tfr1ALT  8387  tfr2ALT  8388  tfr3ALT  8389  on2recsfn  8654  on2recsov  8655  on2ind  8656  on3ind  8657  ordunifi  9281  ordtypelem8  9507  oismo  9522  cantnfcl  9649  leweon  9993  r0weon  9994  ac10ct  10016  dfac12lem2  10126  cflim2  10245  cofsmo  10251  hsmexlem1  10408  smobeth  10568  gruina  10800  ltsopi  10870  finminlem  35108  dnwech  41661  aomclem4  41670  onsupuni  41849  oninfint  41856  epsoon  41873  epirron  41874  oneptr  41875  oaun3lem1  41995