Theorem dfac8b 10149

Description: The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
dfac8b	⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem dfac8b

Dummy variables 𝑤 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardid2 10071	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2		bren 9021	. . 3 ⊢ ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)⟶1-1onto𝐴)
3	1, 2	sylib 221	. 2 ⊢ (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)⟶1-1onto𝐴)
4		sqxpexg 7801	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5		inex2g 5287	. . . . 5 ⊢ ((𝐴 × 𝐴) ∈ V → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V)
6	4, 5	syl 18	. . . 4 ⊢ (𝐴 ∈ dom card → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V)
7		f1ocnv 6867	. . . . . 6 ⊢ (𝑓:(card‘𝐴)⟶1-1onto𝐴 → ◡𝑓:𝐴⟶1-1onto(card‘𝐴))
8		cardon 10062	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
9	8	onordi 6502	. . . . . . . 8 ⊢ Ord (card‘𝐴)
10		ordwe 6401	. . . . . . . 8 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴))
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ E We (card‘𝐴)
12		eqid 2763	. . . . . . . 8 ⊢ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}
13	12	f1owe 7392	. . . . . . 7 ⊢ (◡𝑓:𝐴⟶1-1onto(card‘𝐴) → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴 ↔ E We (card‘𝐴)))
14	11, 13	mpbiri 261	. . . . . 6 ⊢ (◡𝑓:𝐴⟶1-1onto(card‘𝐴) → {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴)
15	7, 14	syl 18	. . . . 5 ⊢ (𝑓:(card‘𝐴)⟶1-1onto𝐴 → {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴)
16		weinxp 5744	. . . . 5 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
17	15, 16	sylib 221	. . . 4 ⊢ (𝑓:(card‘𝐴)⟶1-1onto𝐴 → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
18		weeq1 5645	. . . . 5 ⊢ (𝑥 = ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
19	18	spcegv 3555	. . . 4 ⊢ (({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V → (({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
20	6, 17, 19	syl2im 41	. . 3 ⊢ (𝐴 ∈ dom card → (𝑓:(card‘𝐴)⟶1-1onto𝐴 → ∃𝑥 𝑥 We 𝐴))
21	20	exlimdv 1968	. 2 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:(card‘𝐴)⟶1-1onto𝐴 → ∃𝑥 𝑥 We 𝐴))
22	3, 21	mpd 16	1 ⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴)

This proof depends on syntax axioms:   → wi 4  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092   class class class wbr 5108  {copab 5173   E cep 5557   We wwe 5610   × cxp 5656  ^◡ccnv 5657  dom cdm 5658  Ord word 6387  ⟶1-1ontowf1o 6565  ‘cfv 6566   ≈ cen 9008  cardccrd 10053

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-en 9012  df-card 10057

This theorem is used by:  ween  10153  ac5num  10154  dfac8  10253  numiunnum  37473