Theorem zorng 9326

Description: Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 9329 avoids the Axiom of Choice by assuming that A is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)

Assertion

Ref	Expression
zorng	|- ( ( A e. dom card /\ A. z ( ( z C_ A /\ [ C.] Or z ) -> U. z e. A ) ) -> E. x e. A A. y e. A -. x C. y )

Distinct variable group:   x, y, z, A

Proof of Theorem zorng

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		risset 3062	. . . . . 6 |- ( U. z e. A <-> E. x e. A x = U. z )
2		eqimss2 3658	. . . . . . . . 9 |- ( x = U. z -> U. z C_ x )
3		unissb 4469	. . . . . . . . 9 |- ( U. z C_ x <-> A. u e. z u C_ x )
4	2, 3	sylib 208	. . . . . . . 8 |- ( x = U. z -> A. u e. z u C_ x )
5		vex 3203	. . . . . . . . . . . 12 |- x e. _V
6	5	brrpss 6940	. . . . . . . . . . 11 |- ( u [ C.] x <-> u C. x )
7	6	orbi1i 542	. . . . . . . . . 10 |- ( ( u [ C.] x \/ u = x ) <-> ( u C. x \/ u = x ) )
8		sspss 3706	. . . . . . . . . 10 |- ( u C_ x <-> ( u C. x \/ u = x ) )
9	7, 8	bitr4i 267	. . . . . . . . 9 |- ( ( u [ C.] x \/ u = x ) <-> u C_ x )
10	9	ralbii 2980	. . . . . . . 8 |- ( A. u e. z ( u [ C.] x \/ u = x ) <-> A. u e. z u C_ x )
11	4, 10	sylibr 224	. . . . . . 7 |- ( x = U. z -> A. u e. z ( u [ C.] x \/ u = x ) )
12	11	reximi 3011	. . . . . 6 |- ( E. x e. A x = U. z -> E. x e. A A. u e. z ( u [ C.] x \/ u = x ) )
13	1, 12	sylbi 207	. . . . 5 |- ( U. z e. A -> E. x e. A A. u e. z ( u [ C.] x \/ u = x ) )
14	13	imim2i 16	. . . 4 |- ( ( ( z C_ A /\ [ C.] Or z ) -> U. z e. A ) -> ( ( z C_ A /\ [ C.] Or z ) -> E. x e. A A. u e. z ( u [ C.] x \/ u = x ) ) )
15	14	alimi 1739	. . 3 |- ( A. z ( ( z C_ A /\ [ C.] Or z ) -> U. z e. A ) -> A. z ( ( z C_ A /\ [ C.] Or z ) -> E. x e. A A. u e. z ( u [ C.] x \/ u = x ) ) )
16		porpss 6941	. . . 4 |- [ C.] Po A
17		zorn2g 9325	. . . 4 |- ( ( A e. dom card /\ [ C.] Po A /\ A. z ( ( z C_ A /\ [ C.] Or z ) -> E. x e. A A. u e. z ( u [ C.] x \/ u = x ) ) ) -> E. x e. A A. y e. A -. x [ C.] y )
18	16, 17	mp3an2 1412	. . 3 |- ( ( A e. dom card /\ A. z ( ( z C_ A /\ [ C.] Or z ) -> E. x e. A A. u e. z ( u [ C.] x \/ u = x ) ) ) -> E. x e. A A. y e. A -. x [ C.] y )
19	15, 18	sylan2 491	. 2 |- ( ( A e. dom card /\ A. z ( ( z C_ A /\ [ C.] Or z ) -> U. z e. A ) ) -> E. x e. A A. y e. A -. x [ C.] y )
20		vex 3203	. . . . . 6 |- y e. _V
21	20	brrpss 6940	. . . . 5 |- ( x [ C.] y <-> x C. y )
22	21	notbii 310	. . . 4 |- ( -. x [ C.] y <-> -. x C. y )
23	22	ralbii 2980	. . 3 |- ( A. y e. A -. x [ C.] y <-> A. y e. A -. x C. y )
24	23	rexbii 3041	. 2 |- ( E. x e. A A. y e. A -. x [ C.] y <-> E. x e. A A. y e. A -. x C. y )
25	19, 24	sylib 208	1 |- ( ( A e. dom card /\ A. z ( ( z C_ A /\ [ C.] Or z ) -> U. z e. A ) ) -> E. x e. A A. y e. A -. x C. y )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   C. wpss 3575   U.cuni 4436   class class class wbr 4653   Po wpo 5033   Or wor 5034   dom cdm 5114   [ C.] crpss 6936   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-rpss 6937  df-wrecs 7407  df-recs 7468  df-en 7956  df-card 8765

This theorem is referenced by:  zornn0g  9327  zorn  9329