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Theorem zorng 10529
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 10532 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 3220 . . . . . 6 ( 𝑧𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑧)
2 eqimss2 4036 . . . . . . . . 9 (𝑥 = 𝑧 𝑧𝑥)
3 unissb 4943 . . . . . . . . 9 ( 𝑧𝑥 ↔ ∀𝑢𝑧 𝑢𝑥)
42, 3sylib 217 . . . . . . . 8 (𝑥 = 𝑧 → ∀𝑢𝑧 𝑢𝑥)
5 vex 3465 . . . . . . . . . . . 12 𝑥 ∈ V
65brrpss 7732 . . . . . . . . . . 11 (𝑢 [] 𝑥𝑢𝑥)
76orbi1i 911 . . . . . . . . . 10 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ (𝑢𝑥𝑢 = 𝑥))
8 sspss 4095 . . . . . . . . . 10 (𝑢𝑥 ↔ (𝑢𝑥𝑢 = 𝑥))
97, 8bitr4i 277 . . . . . . . . 9 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ 𝑢𝑥)
109ralbii 3082 . . . . . . . 8 (∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥) ↔ ∀𝑢𝑧 𝑢𝑥)
114, 10sylibr 233 . . . . . . 7 (𝑥 = 𝑧 → ∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1211reximi 3073 . . . . . 6 (∃𝑥𝐴 𝑥 = 𝑧 → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
131, 12sylbi 216 . . . . 5 ( 𝑧𝐴 → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1413imim2i 16 . . . 4 (((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥)))
1514alimi 1805 . . 3 (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥)))
16 porpss 7733 . . . 4 [] Po 𝐴
17 zorn2g 10528 . . . 4 ((𝐴 ∈ dom card ∧ [] Po 𝐴 ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1816, 17mp3an2 1445 . . 3 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1915, 18sylan2 591 . 2 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
20 vex 3465 . . . . . 6 𝑦 ∈ V
2120brrpss 7732 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
2221notbii 319 . . . 4 𝑥 [] 𝑦 ↔ ¬ 𝑥𝑦)
2322ralbii 3082 . . 3 (∀𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦)
2423rexbii 3083 . 2 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
2519, 24sylib 217 1 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845  wal 1531   = wceq 1533  wcel 2098  wral 3050  wrex 3059  wss 3944  wpss 3945   cuni 4909   class class class wbr 5149   Po wpo 5588   Or wor 5589  dom cdm 5678   [] crpss 7728  cardccrd 9960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-rpss 7729  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-en 8965  df-card 9964
This theorem is referenced by:  zornn0g  10530  zorn  10532
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