Theorem dfac8b 9917

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 9841	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2		bren 8874	. . 3 ⊢ ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴)
3	1, 2	sylib 218	. 2 ⊢ (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴)
4		sqxpexg 7683	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5		incom 4154	. . . . . 6 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)})
6		inex1g 5252	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V)
7	5, 6	eqeltrid 2835	. . . . 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 6770	. . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴))
10		cardon 9832	. . . . . . . 8 ⊢ (card‘𝐴) ∈ On
11	10	onordi 6414	. . . . . . 7 ⊢ Ord (card‘𝐴)
12		ordwe 6314	. . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ E We (card‘𝐴)
14		eqid 2731	. . . . . . 7 ⊢ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}
15	14	f1owe 7282	. . . . . 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 5696	. . . . 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 218	. . . 4 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
19		weeq1 5598	. . . . 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 3547	. . . 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 1934	. 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 1780   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   class class class wbr 5086  {copab 5148   E cep 5510   We wwe 5563   × cxp 5609  ^◡ccnv 5610  dom cdm 5611  Ord word 6300  –1-1-onto→wf1o 6475  ‘cfv 6476   ≈ cen 8861  cardccrd 9823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ord 6304  df-on 6305  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-en 8865  df-card 9827

This theorem is referenced by:  ween  9921  ac5num  9922  dfac8  10022  numiunnum  36504