Theorem zorng 10528

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 10531 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 3227	. . . . . 6 ⊢ (∪ 𝑧 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝑧)
2		eqimss2 4039	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑧 → ∪ 𝑧 ⊆ 𝑥)
3		unissb 4942	. . . . . . . . 9 ⊢ (∪ 𝑧 ⊆ 𝑥 ↔ ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
4	2, 3	sylib 217	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑧 → ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
5		vex 3475	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
6	5	brrpss 7731	. . . . . . . . . . 11 ⊢ (𝑢 [⊊] 𝑥 ↔ 𝑢 ⊊ 𝑥)
7	6	orbi1i 912	. . . . . . . . . 10 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
8		sspss 4097	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑥 ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
9	7, 8	bitr4i 278	. . . . . . . . 9 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ 𝑢 ⊆ 𝑥)
10	9	ralbii 3090	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
11	4, 10	sylibr 233	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑧 → ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))
12	11	reximi 3081	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝑧 → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))
13	1, 12	sylbi 216	. . . . 5 ⊢ (∪ 𝑧 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥))
14	13	imim2i 16	. . . 4 ⊢ (((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥)))
15	14	alimi 1806	. . 3 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥)))
16		porpss 7732	. . . 4 ⊢ [⊊] Po 𝐴
17		zorn2g 10527	. . . 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 1446	. . 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 592	. 2 ⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦)
20		vex 3475	. . . . . 6 ⊢ 𝑦 ∈ V
21	20	brrpss 7731	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
22	21	notbii 320	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑦 ↔ ¬ 𝑥 ⊊ 𝑦)
23	22	ralbii 3090	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
24	23	rexbii 3091	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
25	19, 24	sylib 217	1 ⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∀wral 3058  ∃wrex 3067   ⊆ wss 3947   ⊊ wpss 3948  ∪ cuni 4908   class class class wbr 5148   Po wpo 5588   Or wor 5589  dom cdm 5678   [⊊] crpss 7727  cardccrd 9959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  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 6305  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-rpss 7728  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-en 8965  df-card 9963

This theorem is referenced by:  zornn0g  10529  zorn  10531