Theorem zorn2lem6 9206

Description: Lemma for zorn2 9211. (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		zorn2lem.3	. . . . . 6 ⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))
2		zorn2lem.4	. . . . . 6 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
3		zorn2lem.5	. . . . . 6 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
4		zorn2lem.7	. . . . . 6 ⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧}
5	1, 2, 3, 4	zorn2lem5 9205	. . . . 5 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (𝐹 “ 𝑥) ⊆ 𝐴)
6		poss 4961	. . . . 5 ⊢ ((𝐹 “ 𝑥) ⊆ 𝐴 → (𝑅 Po 𝐴 → 𝑅 Po (𝐹 “ 𝑥)))
7	5, 6	syl 17	. . . 4 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (𝑅 Po 𝐴 → 𝑅 Po (𝐹 “ 𝑥)))
8	7	com12 32	. . 3 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Po (𝐹 “ 𝑥)))
9	1	tfr1 7380	. . . . . . . 8 ⊢ 𝐹 Fn On
10		fnfun 5902	. . . . . . . 8 ⊢ (𝐹 Fn On → Fun 𝐹)
11		fvelima 6158	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ (𝐹 “ 𝑥)) → ∃𝑏 ∈ 𝑥 (𝐹‘𝑏) = 𝑠)
12		df-rex 2902	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ 𝑥 (𝐹‘𝑏) = 𝑠 ↔ ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠))
13	11, 12	sylib 207	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ (𝐹 “ 𝑥)) → ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠))
14	13	ex 449	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑠 ∈ (𝐹 “ 𝑥) → ∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠)))
15		fvelima 6158	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑎 ∈ 𝑥 (𝐹‘𝑎) = 𝑟)
16		df-rex 2902	. . . . . . . . . . 11 ⊢ (∃𝑎 ∈ 𝑥 (𝐹‘𝑎) = 𝑟 ↔ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))
17	15, 16	sylib 207	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))
18	17	ex 449	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑟 ∈ (𝐹 “ 𝑥) → ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
19	14, 18	anim12d 584	. . . . . . . 8 ⊢ (Fun 𝐹 → ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟))))
20	9, 10, 19	mp2b 10	. . . . . . 7 ⊢ ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
21		an4 861	. . . . . . . . 9 ⊢ (((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) ↔ ((𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ (𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
22	21	2exbii 1765	. . . . . . . 8 ⊢ (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) ↔ ∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ (𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
23		eeanv 2170	. . . . . . . 8 ⊢ (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ (𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)) ↔ (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
24	22, 23	bitri 263	. . . . . . 7 ⊢ (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) ↔ (∃𝑏(𝑏 ∈ 𝑥 ∧ (𝐹‘𝑏) = 𝑠) ∧ ∃𝑎(𝑎 ∈ 𝑥 ∧ (𝐹‘𝑎) = 𝑟)))
25	20, 24	sylibr 223	. . . . . 6 ⊢ ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → ∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)))
26	4	neeq1i 2846	. . . . . . . . . . 11 ⊢ (𝐻 ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅)
27	26	ralbii 2963	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ ↔ ∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅)
28		imaeq2 5381	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (𝐹 “ 𝑦) = (𝐹 “ 𝑏))
29	28	raleqdv 3121	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑏 → (∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧))
30	29	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧})
31	30	neeq1d 2841	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅))
32	31	rspccv 3279	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → (𝑏 ∈ 𝑥 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅))
33		imaeq2 5381	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑎 → (𝐹 “ 𝑦) = (𝐹 “ 𝑎))
34	33	raleqdv 3121	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧))
35	34	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧})
36	35	neeq1d 2841	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ ↔ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))
37	36	rspccv 3279	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → (𝑎 ∈ 𝑥 → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅))
38	32, 37	anim12d 584	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ≠ ∅ → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)))
39	27, 38	sylbi 206	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → ({𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅ ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)))
40		onelon 5665	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑏 ∈ 𝑥) → 𝑏 ∈ On)
41		onelon 5665	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑎 ∈ 𝑥) → 𝑎 ∈ On)
42	40, 41	anim12dan 878	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥)) → (𝑏 ∈ On ∧ 𝑎 ∈ On))
43	42	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (𝑏 ∈ On ∧ 𝑎 ∈ On)))
44		eloni 5650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ On → Ord 𝑏)
45		eloni 5650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ On → Ord 𝑎)
46		ordtri3or 5672	. . . . . . . . . . . . . . . . . 18 ⊢ ((Ord 𝑏 ∧ Ord 𝑎) → (𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏))
47	44, 45, 46	syl2an 493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏))
48		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧}
49	1, 2, 48	zorn2lem2 9202	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ On ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
50	49	adantll 746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (𝐹‘𝑏)𝑅(𝐹‘𝑎)))
51		breq12 4588	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑏)𝑅(𝐹‘𝑎) ↔ 𝑠𝑅𝑟))
52	51	biimpcd 238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑏)𝑅(𝐹‘𝑎) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑠𝑅𝑟))
53	50, 52	syl6 34	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (𝑏 ∈ 𝑎 → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑠𝑅𝑟)))
54	53	com23 84	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑎)𝑔𝑅𝑧} ≠ ∅)) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑏 ∈ 𝑎 → 𝑠𝑅𝑟)))
55	54	adantrrl 756	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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	55	imp 444	. . . . . . . . . . . . . . . . . 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))
57		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑎 → (𝐹‘𝑏) = (𝐹‘𝑎))
58		eqeq12 2623	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑏) = (𝐹‘𝑎) ↔ 𝑠 = 𝑟))
59	57, 58	syl5ib 233	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑏 = 𝑎 → 𝑠 = 𝑟))
60	59	adantl 481	. . . . . . . . . . . . . . . . . 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 → 𝑠 = 𝑟))
61		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧}
62	1, 2, 61	zorn2lem2 9202	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ On ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
63	62	adantlr 747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
64		breq12 4588	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹‘𝑎) = 𝑟 ∧ (𝐹‘𝑏) = 𝑠) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ↔ 𝑟𝑅𝑠))
65	64	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ↔ 𝑟𝑅𝑠))
66	65	biimpcd 238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑟𝑅𝑠))
67	63, 66	syl6 34	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (𝑎 ∈ 𝑏 → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → 𝑟𝑅𝑠)))
68	67	com23 84	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ On ∧ 𝑎 ∈ On) ∧ (𝑤 We 𝐴 ∧ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑏)𝑔𝑅𝑧} ≠ ∅)) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑎 ∈ 𝑏 → 𝑟𝑅𝑠)))
69	68	adantrrr 757	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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	69	imp 444	. . . . . . . . . . . . . . . . . 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))
71	56, 60, 70	3orim123d 1399	. . . . . . . . . . . . . . . . 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 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
72	47, 71	syl5 33	. . . . . . . . . . . . . . . 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 ∈ On ∧ 𝑎 ∈ On) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
73	72	exp31 628	. . . . . . . . . . . . . . 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)))))
74	73	com4r 92	. . . . . . . . . . . . . 14 ⊢ ((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)))))
75	43, 43, 74	syl6c 68	. . . . . . . . . . . . 13 ⊢ (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	exp4a 631	. . . . . . . . . . . 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))))))
77	76	com3r 85	. . . . . . . . . . 11 ⊢ (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	imp 444	. . . . . . . . . 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)))))
79	78	a2d 29	. . . . . . . . 9 ⊢ ((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)))))
80	39, 79	syl5 33	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) → (((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))))
81	80	imp4b 611	. . . . . . 7 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
82	81	exlimdvv 1849	. . . . . 6 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (∃𝑏∃𝑎((𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥) ∧ ((𝐹‘𝑏) = 𝑠 ∧ (𝐹‘𝑎) = 𝑟)) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
83	25, 82	syl5 33	. . . . 5 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → ((𝑠 ∈ (𝐹 “ 𝑥) ∧ 𝑟 ∈ (𝐹 “ 𝑥)) → (𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
84	83	ralrimivv 2953	. . . 4 ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠))
85	84	a1i 11	. . 3 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
86	8, 85	jcad 554	. 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))))
87		df-so 4960	. 2 ⊢ (𝑅 Or (𝐹 “ 𝑥) ↔ (𝑅 Po (𝐹 “ 𝑥) ∧ ∀𝑠 ∈ (𝐹 “ 𝑥)∀𝑟 ∈ (𝐹 “ 𝑥)(𝑠𝑅𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟𝑅𝑠)))
88	86, 87	syl6ibr 241	1 ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹 “ 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643   Po wpo 4957   Or wor 4958   We wwe 4996  ran crn 5039   “ cima 5041  Ord word 5639  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  ℩crio 6510  recscrecs 7354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-wrecs 7294  df-recs 7355

This theorem is referenced by:  zorn2lem7  9207