Theorem weth 4767

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 4764	. 2 ⊢ ∃y ∈ On ∃f f:y–1-1-onto→A
3		f1ocnv 3692	. . . . . 6 ⊢ (f:y–1-1-onto→A → ◡f:A–1-1-onto→y)
4		eqid 1473	. . . . . . . . 9 ⊢ {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} = {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)}
5	4	f1owe 3896	. . . . . . . 8 ⊢ (◡f:A–1-1-onto→y → (E We y → {⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} We A))
6		weinxp 3228	. . . . . . . . 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 3255	. . . . . . . . . . 11 ⊢ (A × A) ∈ V
8	7	inex2 2712	. . . . . . . . . 10 ⊢ ({⟨z, w⟩∣(◡f ‘z)E(◡f ‘w)} ∩ (A × A)) ∈ V
9		weeq1 2932	. . . . . . . . . 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 1865	. . . . . . . . 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 2953	. . . . . . . 8 ⊢ (y ∈ On → Ord y)
14		ordwe 2956	. . . . . . . 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 1293	. . . 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 1740	. 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 956  ∃wex 978  ∃wrex 1643  Vcvv 1807   ∩ cin 2042   class class class wbr 2614  {copab 2661  Ecep 2825   We wwe 2911  Ord word 2942  Oncon0 2943   × cxp 3163  ^◡ccnv 3164  –1-1-onto→wf1o 3176   ‘cfv 3177

This theorem is referenced by:  zorn2lem7 4774  acdc3 7437  acdc2 7440  acdc5 7443  acdc 7445

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-iso 3194

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