Theorem dfac8b 9953

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 9877	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2		bren 8905	. . 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 7710	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5		incom 4163	. . . . . 6 ⊢ ({⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)})
6		inex1g 5266	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V)
7	5, 6	eqeltrid 2841	. . . . 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 6794	. . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴))
10		cardon 9868	. . . . . . . 8 ⊢ (card‘𝐴) ∈ On
11	10	onordi 6438	. . . . . . 7 ⊢ Ord (card‘𝐴)
12		ordwe 6338	. . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ E We (card‘𝐴)
14		eqid 2737	. . . . . . 7 ⊢ {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}
15	14	f1owe 7309	. . . . . 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 5717	. . . . 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 5619	. . . . 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 3553	. . . 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 1935	. 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 1781   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902   class class class wbr 5100  {copab 5162   E cep 5531   We wwe 5584   × cxp 5630  ^◡ccnv 5631  dom cdm 5632  Ord word 6324  –1-1-onto→wf1o 6499  ‘cfv 6500   ≈ cen 8892  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-en 8896  df-card 9863

This theorem is referenced by:  ween  9957  ac5num  9958  dfac8  10058  numiunnum  36686