Theorem weth 3602

Description: Well-ordering theorem: any set A can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242.

Hypothesis

Ref	Expression
weth.1	⊢ A ∈ V

Assertion

Ref	Expression
weth	⊢ ∃x x We A

Distinct variable group(s):   x,A

Proof of Theorem weth

Step	Hyp	Ref	Expression
1		weth.1	. . 3 ⊢ A ∈ V
2	1	numth 3599	. 2 ⊢ ∃y ∈ On ∃f f:y–1-1-onto→A
3		f1ocnv 2811	. . . . . 6 ⊢ (f:y–1-1-onto→A → ◡f:A–1-1-onto→y)
4		cleqid 1102	. . . . . . . . 9 ⊢ {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} = {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)}
5	4	f1owe 2943	. . . . . . . 8 ⊢ (◡f:A–1-1-onto→y → (E We y → {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} We A))
6		weinxp 2467	. . . . . . . . 9 ⊢ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} We A ↔ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) We A)
7	1, 1	xpex 2488	. . . . . . . . . . 11 ⊢ (A × A) ∈ V
8	7	inex2 1698	. . . . . . . . . 10 ⊢ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) ∈ V
9		weeq1 2189	. . . . . . . . . 10 ⊢ (x = ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) → (x We A ↔ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) We A))
10	8, 9	cla4ev 1401	. . . . . . . . 9 ⊢ (({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) We A → ∃x x We A)
11	6, 10	sylbi 174	. . . . . . . 8 ⊢ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} We A → ∃x x We A)
12	5, 11	syl6 23	. . . . . . 7 ⊢ (◡f:A–1-1-onto→y → (E We y → ∃x x We A))
13		eloni 2209	. . . . . . . 8 ⊢ (y ∈ On → Ord y)
14		ordwe 2212	. . . . . . . 8 ⊢ (Ord y → E We y)
15	13, 14	syl 12	. . . . . . 7 ⊢ (y ∈ On → E We y)
16	12, 15	syl5 22	. . . . . 6 ⊢ (◡f:A–1-1-onto→y → (y ∈ On → ∃x x We A))
17	3, 16	syl 12	. . . . 5 ⊢ (f:y–1-1-onto→A → (y ∈ On → ∃x x We A))
18	17	19.23aiv 952	. . . 4 ⊢ (∃f f:y–1-1-onto→A → (y ∈ On → ∃x x We A))
19	18	com12 13	. . 3 ⊢ (y ∈ On → (∃f f:y–1-1-onto→A → ∃x x We A))
20	19	r19.23aiv 1284	. 2 ⊢ (∃y ∈ On ∃f f:y–1-1-onto→A → ∃x x We A)
21	2, 20	ax-mp 6	1 ⊢ ∃x x We A

Syntax hints:   → wi 2  ∃wex 678   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ∩ cin 1486   class class class wbr 2054  {copab 2055  Ecep 2056   We wwe 2062  Ord word 2198  Oncon0 2199   × cxp 2408  ^◡ccnv 2409  –1-1-onto→wf1o 2421   ‘cfv 2422

This theorem is referenced by:  zornlem7 3609

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439

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