Theorem zorng 10353

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 10356 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 3217	. . . . . 6 ⊢ (∪ 𝑧 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝑧)
2		eqimss2 3988	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑧 → ∪ 𝑧 ⊆ 𝑥)
3		unissb 4886	. . . . . . . . 9 ⊢ (∪ 𝑧 ⊆ 𝑥 ↔ ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
4	2, 3	sylib 217	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑧 → ∀𝑢 ∈ 𝑧 𝑢 ⊆ 𝑥)
5		vex 3445	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
6	5	brrpss 7633	. . . . . . . . . . 11 ⊢ (𝑢 [⊊] 𝑥 ↔ 𝑢 ⊊ 𝑥)
7	6	orbi1i 911	. . . . . . . . . 10 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
8		sspss 4045	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑥 ↔ (𝑢 ⊊ 𝑥 ∨ 𝑢 = 𝑥))
9	7, 8	bitr4i 277	. . . . . . . . 9 ⊢ ((𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥) ↔ 𝑢 ⊆ 𝑥)
10	9	ralbii 3092	. . . . . . . 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 3083	. . . . . 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 1812	. . 3 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∃𝑥 ∈ 𝐴 ∀𝑢 ∈ 𝑧 (𝑢 [⊊] 𝑥 ∨ 𝑢 = 𝑥)))
16		porpss 7634	. . . 4 ⊢ [⊊] Po 𝐴
17		zorn2g 10352	. . . 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 1448	. . 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 3445	. . . . . 6 ⊢ 𝑦 ∈ V
21	20	brrpss 7633	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
22	21	notbii 319	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑦 ↔ ¬ 𝑥 ⊊ 𝑦)
23	22	ralbii 3092	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 [⊊] 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
24	23	rexbii 3093	. 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 844  ∀wal 1538   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ∃wrex 3070   ⊆ wss 3897   ⊊ wpss 3898  ∪ cuni 4851   class class class wbr 5089   Po wpo 5524   Or wor 5525  dom cdm 5614   [⊊] crpss 7629  cardccrd 9784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-rpss 7630  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-en 8797  df-card 9788

This theorem is referenced by:  zornn0g  10354  zorn  10356