Theorem weth 4754

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. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904.

Hypothesis

Ref	Expression
weth.1	⊢ A ∈ V

Assertion

Ref	Expression
weth	⊢ ∃x x We A

Distinct variable group:   x,A

Proof of Theorem weth

Step	Hyp	Ref	Expression
1		weth.1	. . 3 ⊢ A ∈ V
2	1	numth 4751	. 2 ⊢ ∃y ∈ On ∃f f:y–1-1-onto→A
3		f1ocnv 3681	. . . . . 6 ⊢ (f:y–1-1-onto→A → ◡f:A–1-1-onto→y)
4		eqid 1467	. . . . . . . . 9 ⊢ {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} = {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)}
5	4	f1owe 3885	. . . . . . . 8 ⊢ (◡f:A–1-1-onto→y → (E We y → {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} We A))
6		weinxp 3220	. . . . . . . . 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 3247	. . . . . . . . . . 11 ⊢ (A × A) ∈ V
8	7	inex2 2706	. . . . . . . . . 10 ⊢ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) ∈ V
9		weeq1 2924	. . . . . . . . . 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 1859	. . . . . . . . 9 ⊢ (({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) We A → ∃x x We A)
11	6, 10	sylbi 199	. . . . . . . 8 ⊢ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} We A → ∃x x We A)
12	5, 11	syl6 22	. . . . . . 7 ⊢ (◡f:A–1-1-onto→y → (E We y → ∃x x We A))
13		eloni 2945	. . . . . . . 8 ⊢ (y ∈ On → Ord y)
14		ordwe 2948	. . . . . . . 8 ⊢ (Ord y → E We y)
15	13, 14	syl 10	. . . . . . 7 ⊢ (y ∈ On → E We y)
16	12, 15	syl5 21	. . . . . 6 ⊢ (◡f:A–1-1-onto→y → (y ∈ On → ∃x x We A))
17	3, 16	syl 10	. . . . 5 ⊢ (f:y–1-1-onto→A → (y ∈ On → ∃x x We A))
18	17	19.23aiv 1289	. . . 4 ⊢ (∃f f:y–1-1-onto→A → (y ∈ On → ∃x x We A))
19	18	com12 11	. . 3 ⊢ (y ∈ On → (∃f f:y–1-1-onto→A → ∃x x We A))
20	19	r19.23aiv 1734	. 2 ⊢ (∃y ∈ On ∃f f:y–1-1-onto→A → ∃x x We A)
21	2, 20	ax-mp 7	1 ⊢ ∃x x We A

Syntax hints:   → wi 3   ∈ wcel 955  ∃wex 977  ∃wrex 1637  Vcvv 1801   ∩ cin 2035   class class class wbr 2608  {copab 2655  Ecep 2816   We wwe 2903  Ord word 2934  Oncon0 2935   × cxp 3155  ^◡ccnv 3156  –1-1-onto→wf1o 3168   ‘cfv 3169

This theorem is referenced by:  zorn2lem7 4761  acdc3 7424  acdc2 7427  acdc5 7430  acdc 7432

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1451  ax-rep 2682  ax-sep 2692  ax-nul 2699  ax-pow 2731  ax-pr 2768  ax-un 2854  ax-ac 4711

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1374  df-mo 1375  df-clab 1456  df-cleq 1461  df-clel 1464  df-ne 1578  df-ral 1640  df-rex 1641  df-reu 1642  df-rab 1643  df-v 1802  df-sbc 1931  df-dif 2038  df-un 2039  df-in 2040  df-ss 2042  df-nul 2270  df-pw 2391  df-sn 2401  df-pr 2402  df-tp 2404  df-op 2405  df-uni 2493  df-int 2523  df-iun 2557  df-br 2609  df-opab 2656  df-tr 2670  df-eprel 2818  df-id 2821  df-po 2828  df-so 2838  df-fr 2904  df-we 2921  df-ord 2938  df-on 2939  df-suc 2941  df-xp 3171  df-rel 3172  df-cnv 3173  df-co 3174  df-dm 3175  df-rn 3176  df-res 3177  df-ima 3178  df-fun 3179  df-fn 3180  df-f 3181  df-f1 3182  df-fo 3183  df-f1o 3184  df-fv 3185  df-iso 3186

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