Theorem zorn2lem6 10506

Description: Lemma for zorn2 10511. (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 5565	. . . 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 10505	. . . 4 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (𝐹 “ 𝑥) ⊆ 𝐴)
7	1, 6	syl11 34	. . 3 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Po (𝐹 “ 𝑥)))
8	2	tfr1 8387	. . . . . . 7 ⊢ 𝐹 Fn On
9		fnfun 6633	. . . . . . 7 ⊢ (𝐹 Fn On → Fun 𝐹)
10		fvelima 6944	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ (𝐹 “ 𝑥)) → ∃𝑏 ∈ 𝑥 (𝐹‘𝑏) = 𝑠)
11		df-rex 3087	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ 𝑥 (𝐹‘𝑏) = 𝑠 ↔ ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠))
12	10, 11	sylib 221	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ (𝐹 “ 𝑥)) → ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠))
13	12	ex 418	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑠 ∈ (𝐹 “ 𝑥) → ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠)))
14		fvelima 6944	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑎 ∈ 𝑥 (𝐹‘𝑎) = 𝑟)
15		df-rex 3087	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ 𝑥 (𝐹‘𝑎) = 𝑟 ↔ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))
16	14, 15	sylib 221	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))
17	16	ex 418	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑟 ∈ (𝐹 “ 𝑥) → ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
18	13, 17	anim12d 621	. . . . . . 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 669	. . . . . . . 8 ⊢ (((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) ↔ ((𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ (𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
21	20	2exbii 1882	. . . . . . 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 1988	. . . . . . 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 278	. . . . . 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 237	. . . . 5 ⊢ ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)))
25	5	neeq1i 3019	. . . . . . . . . 10 ⊢ (𝐻 ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅)
26	25	ralbii 3108	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ ↔ ∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅)
27		imaeq2 6052	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑏 → (𝐹 “ 𝑦) = (𝐹 “ 𝑏))
28	27	raleqdv 3319	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧))
29	28	rabbidv 3419	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧})
30	29	neeq1d 3014	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅))
31	30	rspccv 3573	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → (𝑏 ∈ 𝑥 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅))
32		imaeq2 6052	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (𝐹 “ 𝑦) = (𝐹 “ 𝑎))
33	32	raleqdv 3319	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧))
34	33	rabbidv 3419	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧})
35	34	neeq1d 3014	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))
36	35	rspccv 3573	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → (𝑎 ∈ 𝑥 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))
37	31, 36	anim12d 621	. . . . . . . . 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 220	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)))
39		onelon 6382	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑏 ∈ 𝑥) → 𝑏 ∈ On)
40		onelon 6382	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑎 ∈ 𝑥) → 𝑎 ∈ On)
41	39, 40	anim12dan 631	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ (𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥)) → (𝑏 ∈ On ∧ 𝑎 ∈ On))
42	41	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (𝑏 ∈ On ∧ 𝑎 ∈ On)))
43		eloni 6367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ On → Ord 𝑏)
44		eloni 6367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ On → Ord 𝑎)
45		ordtri3or 6390	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝑏 ∧ Ord 𝑎) → (𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏))
46	43, 44, 45	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏))
47		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧}
48	2, 3, 47	zorn2lem2 10502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ On ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
49	48	adantll 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
50		breq12 5108	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ↔ 𝑠𝑅𝑟))
51	50	biimpcd 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑏)𝑅(𝐹‘𝑎) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑠𝑅𝑟))
52	49, 51	syl6 36	. . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 6879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑎 → (𝐹‘𝑏) = (𝐹‘𝑎))
57		eqeq12 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑏) = (𝐹‘𝑎) ↔ 𝑠 = 𝑟))
58	56, 57	imbitrid 247	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑏 = 𝑎 → 𝑠 = 𝑟))
59	58	adantl 487	. . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧}
61	2, 3, 60	zorn2lem2 10502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ On ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
62	61	adantlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
63		breq12 5108	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹‘𝑎) = 𝑟 ∧ (𝐹‘𝑏) = 𝑠) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ↔ 𝑟𝑅𝑠))
64	63	ancoms 464	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ↔ 𝑟𝑅𝑠))
65	64	biimpcd 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑟𝑅𝑠))
66	62, 65	syl6 36	. . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 1472	. . . . . . . . . . . . . . . 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 35	. . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . 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 95	. . . . . . . . . . . . 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 71	. . . . . . . . . . . 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 437	. . . . . . . . . . 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 88	. . . . . . . . . 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 412	. . . . . . . . 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 30	. . . . . . . 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 35	. . . . . . 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 427	. . . . . 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 1967	. . . . 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 35	. . . 4 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
83	82	ralrimivv 3203	. . 3 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠))
84	7, 83	jca2 523	. 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 5564	. 2 ⊢ (𝑅 Or (𝐹 “ 𝑥) ↔ (𝑅 Po (𝐹 “ 𝑥) ∧ ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
86	84, 85	imbitrrdi 255	1 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹 “ 𝑥)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   ↦ cmpt 5186   Po wpo 5561   Or wor 5562   We wwe 5607  ran crn 5656   “ cima 5658  Ord word 6356  Oncon0 6357  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533  ℩crio 7370  recscrecs 8360

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  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 6299  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361

This theorem is used by:  zorn2lem7  10507