Theorem zorn2 9328

Description: Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set A (with an ordering relation R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 9318 through zorn2lem7 9324; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 9324. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
zornn0.1	|- A e. _V

Assertion

Ref	Expression
zorn2	|- ( ( R Po A /\ A. w ( ( w C_ A /\ R Or w ) -> E. x e. A A. z e. w ( z R x \/ z = x ) ) ) -> E. x e. A A. y e. A -. x R y )

Distinct variable groups:   x, w, y, z, A   w, R, x, y, z

Proof of Theorem zorn2

Step	Hyp	Ref	Expression
1		zornn0.1	. . 3 |- A e. _V
2		numth3 9292	. . 3 |- ( A e. _V -> A e. dom card )
3	1, 2	ax-mp 5	. 2 |- A e. dom card
4		zorn2g 9325	. 2 |- ( ( A e. dom card /\ R Po A /\ A. w ( ( w C_ A /\ R Or w ) -> E. x e. A A. z e. w ( z R x \/ z = x ) ) ) -> E. x e. A A. y e. A -. x R y )
5	3, 4	mp3an1 1411	1 |- ( ( R Po A /\ A. w ( ( w C_ A /\ R Or w ) -> E. x e. A A. z e. w ( z R x \/ z = x ) ) ) -> E. x e. A A. y e. A -. x R y )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Po wpo 5033   Or wor 5034   dom cdm 5114   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  ax-ac2 9285

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-wrecs 7407  df-recs 7468  df-en 7956  df-card 8765  df-ac 8939

This theorem is referenced by: (None)