Theorem zorn2lem6 10454

Description: Lemma for zorn2 10459. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
zorn2lem.3	⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))
zorn2lem.4	⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
zorn2lem.5	⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
zorn2lem.7	⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧}

Assertion

Ref	Expression
zorn2lem6	⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹 “ 𝑥)))

Distinct variable groups:   𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧,𝐴   𝐷,𝑓,𝑢,𝑣,𝑦   𝑓,𝐹,𝑔,𝑢,𝑣,𝑥,𝑦,𝑧   𝑅,𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑣,𝐶   𝑥,𝐻,𝑢,𝑣,𝑓

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓,𝑔)   𝐷(𝑥,𝑧,𝑤,𝑔)   𝐹(𝑤)   𝐻(𝑦,𝑧,𝑤,𝑔)

Proof of Theorem zorn2lem6

Dummy variables 𝑎 𝑏 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poss 5548	. . . 4 ⊢ ((𝐹 “ 𝑥) ⊆ 𝐴 → (𝑅 Po 𝐴 → 𝑅 Po (𝐹 “ 𝑥)))
2		zorn2lem.3	. . . . 5 ⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))
3		zorn2lem.4	. . . . 5 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
4		zorn2lem.5	. . . . 5 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
5		zorn2lem.7	. . . . 5 ⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧}
6	2, 3, 4, 5	zorn2lem5 10453	. . . 4 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (𝐹 “ 𝑥) ⊆ 𝐴)
7	1, 6	syl11 33	. . 3 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Po (𝐹 “ 𝑥)))
8	2	tfr1 8365	. . . . . . 7 ⊢ 𝐹 Fn On
9		fnfun 6618	. . . . . . 7 ⊢ (𝐹 Fn On → Fun 𝐹)
10		fvelima 6926	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ (𝐹 “ 𝑥)) → ∃𝑏 ∈ 𝑥 (𝐹‘𝑏) = 𝑠)
11		df-rex 3054	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ 𝑥 (𝐹‘𝑏) = 𝑠 ↔ ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠))
12	10, 11	sylib 218	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ (𝐹 “ 𝑥)) → ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠))
13	12	ex 412	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑠 ∈ (𝐹 “ 𝑥) → ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠)))
14		fvelima 6926	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑎 ∈ 𝑥 (𝐹‘𝑎) = 𝑟)
15		df-rex 3054	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ 𝑥 (𝐹‘𝑎) = 𝑟 ↔ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))
16	14, 15	sylib 218	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))
17	16	ex 412	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑟 ∈ (𝐹 “ 𝑥) → ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
18	13, 17	anim12d 609	. . . . . . 7 ⊢ (Fun 𝐹 → ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))))
19	8, 9, 18	mp2b 10	. . . . . 6 ⊢ ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
20		an4 656	. . . . . . . 8 ⊢ (((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) ↔ ((𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ (𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
21	20	2exbii 1849	. . . . . . 7 ⊢ (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) ↔ ∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ (𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
22		exdistrv 1955	. . . . . . 7 ⊢ (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ (𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)) ↔ (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
23	21, 22	bitri 275	. . . . . 6 ⊢ (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) ↔ (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
24	19, 23	sylibr 234	. . . . 5 ⊢ ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)))
25	5	neeq1i 2989	. . . . . . . . . 10 ⊢ (𝐻 ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅)
26	25	ralbii 3075	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ ↔ ∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅)
27		imaeq2 6027	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑏 → (𝐹 “ 𝑦) = (𝐹 “ 𝑏))
28	27	raleqdv 3299	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧))
29	28	rabbidv 3413	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧})
30	29	neeq1d 2984	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅))
31	30	rspccv 3585	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → (𝑏 ∈ 𝑥 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅))
32		imaeq2 6027	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (𝐹 “ 𝑦) = (𝐹 “ 𝑎))
33	32	raleqdv 3299	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧))
34	33	rabbidv 3413	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧})
35	34	neeq1d 2984	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))
36	35	rspccv 3585	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → (𝑎 ∈ 𝑥 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))
37	31, 36	anim12d 609	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)))
38	26, 37	sylbi 217	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)))
39		onelon 6357	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑏 ∈ 𝑥) → 𝑏 ∈ On)
40		onelon 6357	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑎 ∈ 𝑥) → 𝑎 ∈ On)
41	39, 40	anim12dan 619	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ (𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥)) → (𝑏 ∈ On ∧ 𝑎 ∈ On))
42	41	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (𝑏 ∈ On ∧ 𝑎 ∈ On)))
43		eloni 6342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ On → Ord 𝑏)
44		eloni 6342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ On → Ord 𝑎)
45		ordtri3or 6364	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑏 ∧ Ord 𝑎) → (𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏))
46	43, 44, 45	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏))
47		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧}
48	2, 3, 47	zorn2lem2 10450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ On ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
49	48	adantll 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
50		breq12 5112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ↔ 𝑠𝑅𝑟))
51	50	biimpcd 249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑏)𝑅(𝐹‘𝑎) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑠𝑅𝑟))
52	49, 51	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑠𝑅𝑟)))
53	52	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑏 ∈ 𝑎 → 𝑠𝑅𝑟)))
54	53	adantrrl 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑏 ∈ 𝑎 → 𝑠𝑅𝑟)))
55	54	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → (𝑏 ∈ 𝑎 → 𝑠𝑅𝑟))
56		fveq2 6858	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑎 → (𝐹‘𝑏) = (𝐹‘𝑎))
57		eqeq12 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑏) = (𝐹‘𝑎) ↔ 𝑠 = 𝑟))
58	56, 57	imbitrid 244	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑏 = 𝑎 → 𝑠 = 𝑟))
59	58	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → (𝑏 = 𝑎 → 𝑠 = 𝑟))
60		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧}
61	2, 3, 60	zorn2lem2 10450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ On ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
62	61	adantlr 715	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
63		breq12 5112	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹‘𝑎) = 𝑟 ∧ (𝐹‘𝑏) = 𝑠) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ↔ 𝑟𝑅𝑠))
64	63	ancoms 458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ↔ 𝑟𝑅𝑠))
65	64	biimpcd 249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑟𝑅𝑠))
66	62, 65	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑟𝑅𝑠)))
67	66	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑎 ∈ 𝑏 → 𝑟𝑅𝑠)))
68	67	adantrrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑎 ∈ 𝑏 → 𝑟𝑅𝑠)))
69	68	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → (𝑎 ∈ 𝑏 → 𝑟𝑅𝑠))
70	55, 59, 69	3orim123d 1446	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → ((𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
71	46, 70	syl5 34	. . . . . . . . . . . . . . 15 ⊢ ((((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
72	71	exp31 419	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → ((𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))))
73	72	com4r 94	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → ((𝑏 ∈ On ∧ 𝑎 ∈ On) → ((𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))))
74	42, 42, 73	syl6c 70	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → ((𝑤 We 𝐴 ∧ ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))))
75	74	exp4a 431	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (𝑤 We 𝐴 → (({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠))))))
76	75	com3r 87	. . . . . . . . . 10 ⊢ (𝑤 We 𝐴 → (𝑥 ∈ On → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠))))))
77	76	imp 406	. . . . . . . . 9 ⊢ ((𝑤 We 𝐴 ∧ 𝑥 ∈ On) → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))))
78	77	a2d 29	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ 𝑥 ∈ On) → (((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))))
79	38, 78	syl5 34	. . . . . . 7 ⊢ ((𝑤 We 𝐴 ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))))
80	79	imp4b 421	. . . . . 6 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
81	80	exlimdvv 1934	. . . . 5 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
82	24, 81	syl5 34	. . . 4 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
83	82	ralrimivv 3178	. . 3 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠))
84	7, 83	jca2 513	. 2 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (𝑅 Po (𝐹 “ 𝑥) ∧ ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠))))
85		df-so 5547	. 2 ⊢ (𝑅 Or (𝐹 “ 𝑥) ↔ (𝑅 Po (𝐹 “ 𝑥) ∧ ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
86	84, 85	imbitrrdi 252	1 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹 “ 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ⊆ wss 3914  ∅c0 4296   class class class wbr 5107   ↦ cmpt 5188   Po wpo 5544   Or wor 5545   We wwe 5590  ran crn 5639   “ cima 5641  Ord word 6331  Oncon0 6332  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  ℩crio 7343  recscrecs 8339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340

This theorem is referenced by:  zorn2lem7  10455