Theorem zorng 10609

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 10612 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 3241	. . . . . 6 ⊢ (∪ 𝑧 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝑧)
2		eqimss2 3972	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑧 → ∪ 𝑧 ⊆ 𝑥)
3		unissb 4904	. . . . . . . . 9 ⊢ (∪ 𝑧 ⊆ 𝑥 ↔ ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
4	2, 3	sylib 221	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑧 → ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
5		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
6	5	brrpss 7766	. . . . . . . . . . 11 ⊢ (𝑢 [⊊] 𝑥 ↔ 𝑢 ⊊ 𝑥)
7	6	orbi1i 927	. . . . . . . . . 10 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
8		sspss 4032	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑥 ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
9	7, 8	bitr4i 281	. . . . . . . . 9 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ 𝑢 ⊆ 𝑥)
10	9	ralbii 3112	. . . . . . . 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 3104	. . . . . 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 7767	. . . 4 ⊢ [⊊] Po 𝐴
17		zorn2g 10608	. . . 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 1479	. . 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 3458	. . . . . 6 ⊢ 𝑦 ∈ V
21	20	brrpss 7766	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
22	21	notbii 323	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑦 ↔ ¬ 𝑥 ⊊ 𝑦)
23	22	ralbii 3112	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
24	23	rexbii 3113	. 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 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ⊊ wpss 4016  ∪ cuni 4870   class class class wbr 5106   Po wpo 5561   Or wor 5562  dom cdm 5655   [⊊] crpss 7762  cardccrd 10043

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-rpss 7763  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-en 9002  df-card 10047

This theorem is used by:  zornn0g  10610  zorn  10612