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Theorem zorng 10441
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 10444 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 3222 . . . . . 6 ( 𝑧𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑧)
2 eqimss2 4002 . . . . . . . . 9 (𝑥 = 𝑧 𝑧𝑥)
3 unissb 4901 . . . . . . . . 9 ( 𝑧𝑥 ↔ ∀𝑢𝑧 𝑢𝑥)
42, 3sylib 217 . . . . . . . 8 (𝑥 = 𝑧 → ∀𝑢𝑧 𝑢𝑥)
5 vex 3450 . . . . . . . . . . . 12 𝑥 ∈ V
65brrpss 7664 . . . . . . . . . . 11 (𝑢 [] 𝑥𝑢𝑥)
76orbi1i 913 . . . . . . . . . 10 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ (𝑢𝑥𝑢 = 𝑥))
8 sspss 4060 . . . . . . . . . 10 (𝑢𝑥 ↔ (𝑢𝑥𝑢 = 𝑥))
97, 8bitr4i 278 . . . . . . . . 9 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ 𝑢𝑥)
109ralbii 3097 . . . . . . . 8 (∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥) ↔ ∀𝑢𝑧 𝑢𝑥)
114, 10sylibr 233 . . . . . . 7 (𝑥 = 𝑧 → ∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1211reximi 3088 . . . . . 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 1814 . . 3 (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥)))
16 porpss 7665 . . . 4 [] Po 𝐴
17 zorn2g 10440 . . . 4 ((𝐴 ∈ dom card ∧ [] Po 𝐴 ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1816, 17mp3an2 1450 . . 3 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1915, 18sylan2 594 . 2 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
20 vex 3450 . . . . . 6 𝑦 ∈ V
2120brrpss 7664 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
2221notbii 320 . . . 4 𝑥 [] 𝑦 ↔ ¬ 𝑥𝑦)
2322ralbii 3097 . . 3 (∀𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦)
2423rexbii 3098 . 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 397  wo 846  wal 1540   = wceq 1542  wcel 2107  wral 3065  wrex 3074  wss 3911  wpss 3912   cuni 4866   class class class wbr 5106   Po wpo 5544   Or wor 5545  dom cdm 5634   [] crpss 7660  cardccrd 9872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-rpss 7661  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-en 8885  df-card 9876
This theorem is referenced by:  zornn0g  10442  zorn  10444
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