Theorem dfac8b 10025

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 9947	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2		bren 8948	. . 3 ⊢ ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴)
3	1, 2	sylib 217	. 2 ⊢ (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴)
4		sqxpexg 7738	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5		incom 4196	. . . . . 6 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)})
6		inex1g 5312	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V)
7	5, 6	eqeltrid 2831	. . . . 5 ⊢ ((𝐴 × 𝐴) ∈ V → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V)
8	4, 7	syl 17	. . . 4 ⊢ (𝐴 ∈ dom card → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V)
9		f1ocnv 6838	. . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴))
10		cardon 9938	. . . . . . . 8 ⊢ (card‘𝐴) ∈ On
11	10	onordi 6468	. . . . . . 7 ⊢ Ord (card‘𝐴)
12		ordwe 6370	. . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ E We (card‘𝐴)
14		eqid 2726	. . . . . . 7 ⊢ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}
15	14	f1owe 7345	. . . . . 6 ⊢ (◡𝑓:𝐴–1-1-onto→(card‘𝐴) → ( E We (card‘𝐴) → {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴))
16	9, 13, 15	mpisyl 21	. . . . 5 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴)
17		weinxp 5753	. . . . 5 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
18	16, 17	sylib 217	. . . 4 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
19		weeq1 5657	. . . . 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))
20	19	spcegv 3581	. . . 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))
21	8, 18, 20	syl2im 40	. . 3 ⊢ (𝐴 ∈ dom card → (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ∃𝑥 𝑥 We 𝐴))
22	21	exlimdv 1928	. 2 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴 → ∃𝑥 𝑥 We 𝐴))
23	3, 22	mpd 15	1 ⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴)

Syntax hints:   → wi 4  ∃wex 1773   ∈ wcel 2098  Vcvv 3468   ∩ cin 3942   class class class wbr 5141  {copab 5203   E cep 5572   We wwe 5623   × cxp 5667  ^◡ccnv 5668  dom cdm 5669  Ord word 6356  –1-1-onto→wf1o 6535  ‘cfv 6536   ≈ cen 8935  cardccrd 9929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-en 8939  df-card 9933

This theorem is referenced by:  ween  10029  ac5num  10030  dfac8  10129