Theorem zorng 10506

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 10509 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((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 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 3243	. . . . . 6 ⊢ (∪ 𝑧 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝑧)
2		eqimss2 3999	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑧 → ∪ 𝑧 ⊆ 𝑥)
3		unissb 4911	. . . . . . . . 9 ⊢ (∪ 𝑧 ⊆ 𝑥 ↔ ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
4	2, 3	sylib 221	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑧 → ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
5		vex 3462	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
6	5	brrpss 7736	. . . . . . . . . . 11 ⊢ (𝑢 [⊊] 𝑥 ↔ 𝑢 ⊊ 𝑥)
7	6	orbi1i 927	. . . . . . . . . 10 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
8		sspss 4059	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑥 ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
9	7, 8	bitr4i 281	. . . . . . . . 9 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ 𝑢 ⊆ 𝑥)
10	9	ralbii 3114	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
11	4, 10	sylibr 237	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑧 → ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))
12	11	reximi 3106	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝑧 → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))
13	1, 12	sylbi 220	. . . . 5 ⊢ (∪ 𝑧 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))
14	13	imim2i 17	. . . 4 ⊢ (((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥)))
15	14	alimi 1844	. . 3 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥)))
16		porpss 7737	. . . 4 ⊢ [⊊] Po 𝐴
17		zorn2g 10505	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ [⊊] Po 𝐴 ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦)
18	16, 17	mp3an2 1478	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦)
19	15, 18	sylan2 605	. 2 ⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦)
20		vex 3462	. . . . . 6 ⊢ 𝑦 ∈ V
21	20	brrpss 7736	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
22	21	notbii 323	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑦 ↔ ¬ 𝑥 ⊊ 𝑦)
23	22	ralbii 3114	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
24	23	rexbii 3115	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
25	19, 24	sylib 221	1 ⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   ⊆ wss 3908   ⊊ wpss 3909  ∪ cuni 4877   class class class wbr 5114   Po wpo 5572   Or wor 5573  dom cdm 5666   [⊊] crpss 7732  cardccrd 9940

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-rpss 7733  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-en 8953  df-card 9944

This theorem is used by:  zornn0g  10507  zorn  10509