Theorem weth 4804

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 e. V

Assertion

Ref	Expression
weth	|- E.x x We A

Distinct variable group:   x,A

Proof of Theorem weth

Step	Hyp	Ref	Expression
1		weth.1	. . 3 |- A e. V
2	1	numth 4801	. 2 |- E.y e. On E.f f:y-1-1-onto->A
3		f1ocnv 3708	. . . . . 6 |- (f:y-1-1-onto->A -> `'f:A-1-1-onto->y)
4		eqid 1478	. . . . . . . . 9 |- {<.z, w>. | (`'f` z)E(`'f` w)} = {<.z, w>. | (`'f` z)E(`'f` w)}
5	4	f1owe 3912	. . . . . . . 8 |- (`'f:A-1-1-onto->y -> (E We y -> {<.z, w>. | (`'f` z)E(`'f` w)} We A))
6		weinxp 3240	. . . . . . . . 9 |- ({<.z, w>. | (`'f` z)E(`'f` w)} We A <-> ({<.z, w>. | (`'f` z)E(`'f` w)} i^i (A X. A)) We A)
7	1, 1	xpex 3267	. . . . . . . . . . 11 |- (A X. A) e. V
8	7	inex2 2723	. . . . . . . . . 10 |- ({<.z, w>. | (`'f` z)E(`'f` w)} i^i (A X. A)) e. V
9		weeq1 2944	. . . . . . . . . 10 |- (x = ({<.z, w>. | (`'f` z)E(`'f` w)} i^i (A X. A)) -> (x We A <-> ({<.z, w>. | (`'f` z)E(`'f` w)} i^i (A X. A)) We A))
10	8, 9	cla4ev 1872	. . . . . . . . 9 |- (({<.z, w>. | (`'f` z)E(`'f` w)} i^i (A X. A)) We A -> E.x x We A)
11	6, 10	sylbi 199	. . . . . . . 8 |- ({<.z, w>. | (`'f` z)E(`'f` w)} We A -> E.x x We A)
12	5, 11	syl6 22	. . . . . . 7 |- (`'f:A-1-1-onto->y -> (E We y -> E.x x We A))
13		eloni 2965	. . . . . . . 8 |- (y e. On -> Ord y)
14		ordwe 2968	. . . . . . . 8 |- (Ord y -> E We y)
15	13, 14	syl 10	. . . . . . 7 |- (y e. On -> E We y)
16	12, 15	syl5 21	. . . . . 6 |- (`'f:A-1-1-onto->y -> (y e. On -> E.x x We A))
17	3, 16	syl 10	. . . . 5 |- (f:y-1-1-onto->A -> (y e. On -> E.x x We A))
18	17	19.23aiv 1297	. . . 4 |- (E.f f:y-1-1-onto->A -> (y e. On -> E.x x We A))
19	18	com12 11	. . 3 |- (y e. On -> (E.f f:y-1-1-onto->A -> E.x x We A))
20	19	r19.23aiv 1746	. 2 |- (E.y e. On E.f f:y-1-1-onto->A -> E.x x We A)
21	2, 20	ax-mp 7	1 |- E.x x We A

Syntax hints:   -> wi 3   e. wcel 960  E.wex 982  E.wrex 1649  Vcvv 1814   i^i cin 2050   class class class wbr 2625  {copab 2672  Ecep 2837   We wwe 2923  Ord word 2954  Oncon0 2955   X. cxp 3175  `'ccnv 3176  -1-1-onto->wf1o 3188  ` cfv 3189

This theorem is referenced by:  zorn2lem7 4811  acdc3 7495  acdc2 7498  acdc5 7501  acdc 7503

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873  ax-ac 4761

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-int 2539  df-iun 2573  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-id 2842  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-suc 2961  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-f1 3202  df-fo 3203  df-f1o 3204  df-fv 3205  df-iso 3206

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