Theorem dfac8b 10026

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 9948	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2		bren 8949	. . 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 7742	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5		incom 4202	. . . . . 6 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)})
6		inex1g 5320	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V)
7	5, 6	eqeltrid 2838	. . . . 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 6846	. . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴))
10		cardon 9939	. . . . . . . 8 ⊢ (card‘𝐴) ∈ On
11	10	onordi 6476	. . . . . . 7 ⊢ Ord (card‘𝐴)
12		ordwe 6378	. . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ E We (card‘𝐴)
14		eqid 2733	. . . . . . 7 ⊢ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}
15	14	f1owe 7350	. . . . . 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 5761	. . . . 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 5665	. . . . 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 3588	. . . 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 1937	. 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 1782   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948   class class class wbr 5149  {copab 5211   E cep 5580   We wwe 5631   × cxp 5675  ^◡ccnv 5676  dom cdm 5677  Ord word 6364  –1-1-onto→wf1o 6543  ‘cfv 6544   ≈ cen 8936  cardccrd 9930

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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-en 8940  df-card 9934

This theorem is referenced by:  ween  10030  ac5num  10031  dfac8  10130