Theorem zorng 10495

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 10498 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 3230	. . . . . 6 ⊢ (∪ 𝑧 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝑧)
2		eqimss2 4040	. . . . . . . . 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 3478	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
6	5	brrpss 7712	. . . . . . . . . . 11 ⊢ (𝑢 [⊊] 𝑥 ↔ 𝑢 ⊊ 𝑥)
7	6	orbi1i 912	. . . . . . . . . 10 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
8		sspss 4098	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑥 ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
9	7, 8	bitr4i 277	. . . . . . . . 9 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ 𝑢 ⊆ 𝑥)
10	9	ralbii 3093	. . . . . . . 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 3084	. . . . . 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 1813	. . 3 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥)))
16		porpss 7713	. . . 4 ⊢ [⊊] Po 𝐴
17		zorn2g 10494	. . . 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 1449	. . 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 593	. 2 ⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦)
20		vex 3478	. . . . . 6 ⊢ 𝑦 ∈ V
21	20	brrpss 7712	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
22	21	notbii 319	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑦 ↔ ¬ 𝑥 ⊊ 𝑦)
23	22	ralbii 3093	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
24	23	rexbii 3094	. 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 396   ∨ wo 845  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   ⊆ wss 3947   ⊊ wpss 3948  ∪ cuni 4907   class class class wbr 5147   Po wpo 5585   Or wor 5586  dom cdm 5675   [⊊] crpss 7708  cardccrd 9926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-rpss 7709  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-en 8936  df-card 9930

This theorem is referenced by:  zornn0g  10496  zorn  10498