MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zorng Structured version   Visualization version   GIF version

Theorem zorng 9579
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 9582 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 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem zorng
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 risset 3209 . . . . . 6 ( 𝑧𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑧)
2 eqimss2 3818 . . . . . . . . 9 (𝑥 = 𝑧 𝑧𝑥)
3 unissb 4627 . . . . . . . . 9 ( 𝑧𝑥 ↔ ∀𝑢𝑧 𝑢𝑥)
42, 3sylib 209 . . . . . . . 8 (𝑥 = 𝑧 → ∀𝑢𝑧 𝑢𝑥)
5 vex 3353 . . . . . . . . . . . 12 𝑥 ∈ V
65brrpss 7138 . . . . . . . . . . 11 (𝑢 [] 𝑥𝑢𝑥)
76orbi1i 937 . . . . . . . . . 10 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ (𝑢𝑥𝑢 = 𝑥))
8 sspss 3867 . . . . . . . . . 10 (𝑢𝑥 ↔ (𝑢𝑥𝑢 = 𝑥))
97, 8bitr4i 269 . . . . . . . . 9 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ 𝑢𝑥)
109ralbii 3127 . . . . . . . 8 (∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥) ↔ ∀𝑢𝑧 𝑢𝑥)
114, 10sylibr 225 . . . . . . 7 (𝑥 = 𝑧 → ∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1211reximi 3157 . . . . . 6 (∃𝑥𝐴 𝑥 = 𝑧 → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
131, 12sylbi 208 . . . . 5 ( 𝑧𝐴 → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1413imim2i 16 . . . 4 (((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥)))
1514alimi 1906 . . 3 (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥)))
16 porpss 7139 . . . 4 [] Po 𝐴
17 zorn2g 9578 . . . 4 ((𝐴 ∈ dom card ∧ [] Po 𝐴 ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1816, 17mp3an2 1573 . . 3 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1915, 18sylan2 586 . 2 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
20 vex 3353 . . . . . 6 𝑦 ∈ V
2120brrpss 7138 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
2221notbii 311 . . . 4 𝑥 [] 𝑦 ↔ ¬ 𝑥𝑦)
2322ralbii 3127 . . 3 (∀𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦)
2423rexbii 3188 . 2 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
2519, 24sylib 209 1 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873  wal 1650   = wceq 1652  wcel 2155  wral 3055  wrex 3056  wss 3732  wpss 3733   cuni 4594   class class class wbr 4809   Po wpo 5196   Or wor 5197  dom cdm 5277   [] crpss 7134  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-rpss 7135  df-wrecs 7610  df-recs 7672  df-en 8161  df-card 9016
This theorem is referenced by:  zornn0g  9580  zorn  9582
  Copyright terms: Public domain W3C validator