Theorem acexmidlemcase 5917

Description: Lemma for acexmid 5921. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle).

The cases are (1) the choice function evaluated at 𝐴 equals {∅}, (2) the choice function evaluated at 𝐵 equals ∅, and (3) the choice function evaluated at 𝐴 equals ∅ and the choice function evaluated at 𝐵 equals {∅}.

Because of the way we represent the choice function 𝑦, the choice function evaluated at 𝐴 is (℩𝑣 ∈ 𝐴∃𝑢 ∈ 𝑦(𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) and the choice function evaluated at 𝐵 is (℩𝑣 ∈ 𝐵∃𝑢 ∈ 𝑦(𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)). Other than the difference in notation these work just as (𝑦‘𝐴) and (𝑦‘𝐵) would if 𝑦 were a function as defined by df-fun 5260.

Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at 𝐴 equals {∅}, then {∅} ∈ 𝐴 and likewise for 𝐵.

(Contributed by Jim Kingdon, 7-Aug-2019.)

Hypotheses

Ref	Expression
acexmidlem.a	⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
acexmidlem.b	⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
acexmidlem.c	⊢ 𝐶 = {𝐴, 𝐵}

Assertion

Ref	Expression
acexmidlemcase	⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑣,𝑢   𝜑,𝑥,𝑦,𝑧,𝑣,𝑢

Proof of Theorem acexmidlemcase

Step	Hyp	Ref	Expression
1		acexmidlem.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
2		onsucelsucexmidlem 4565	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
3	1, 2	eqeltri 2269	. . . . . . . . . . . . 13 ⊢ 𝐴 ∈ On
4		prid1g 3726	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → 𝐴 ∈ {𝐴, 𝐵})
5	3, 4	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ {𝐴, 𝐵}
6		acexmidlem.c	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝐴, 𝐵}
7	5, 6	eleqtrri 2272	. . . . . . . . . . 11 ⊢ 𝐴 ∈ 𝐶
8		eleq1 2259	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐴 → (𝑧 ∈ 𝑢 ↔ 𝐴 ∈ 𝑢))
9	8	anbi1d 465	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐴 → ((𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
10	9	rexbidv 2498	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐴 → (∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
11	10	reueqd 2707	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐴 → (∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃!𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
12	11	rspcv 2864	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐶 → (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ∃!𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
13	7, 12	ax-mp 5	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ∃!𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
14		riotacl 5892	. . . . . . . . . 10 ⊢ (∃!𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐴)
15	13, 14	syl 14	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐴)
16		elrabi 2917	. . . . . . . . . 10 ⊢ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {∅, {∅}})
17	16, 1	eleq2s 2291	. . . . . . . . 9 ⊢ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐴 → (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {∅, {∅}})
18		elpri 3645	. . . . . . . . 9 ⊢ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {∅, {∅}} → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
19	15, 17, 18	3syl 17	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
20		eleq1 2259	. . . . . . . . . 10 ⊢ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅} → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐴 ↔ {∅} ∈ 𝐴))
21	15, 20	syl5ibcom 155	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅} → {∅} ∈ 𝐴))
22	21	orim2d 789	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ {∅} ∈ 𝐴)))
23	19, 22	mpd 13	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ {∅} ∈ 𝐴))
24		acexmidlem.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25		pp0ex 4222	. . . . . . . . . . . . . . 15 ⊢ {∅, {∅}} ∈ V
26	25	rabex 4177	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ∈ V
27	24, 26	eqeltri 2269	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ V
28	27	prid2 3729	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ {𝐴, 𝐵}
29	28, 6	eleqtrri 2272	. . . . . . . . . . 11 ⊢ 𝐵 ∈ 𝐶
30		eleq1 2259	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
31	30	anbi1d 465	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐵 → ((𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
32	31	rexbidv 2498	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐵 → (∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
33	32	reueqd 2707	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → (∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃!𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
34	33	rspcv 2864	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝐶 → (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ∃!𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
35	29, 34	ax-mp 5	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ∃!𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
36		riotacl 5892	. . . . . . . . . 10 ⊢ (∃!𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐵)
37	35, 36	syl 14	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐵)
38		elrabi 2917	. . . . . . . . . 10 ⊢ ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {∅, {∅}})
39	38, 24	eleq2s 2291	. . . . . . . . 9 ⊢ ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐵 → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {∅, {∅}})
40		elpri 3645	. . . . . . . . 9 ⊢ ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ {∅, {∅}} → ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
41	37, 39, 40	3syl 17	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
42		eleq1 2259	. . . . . . . . . 10 ⊢ ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ → ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ∈ 𝐵 ↔ ∅ ∈ 𝐵))
43	37, 42	syl5ibcom 155	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ → ∅ ∈ 𝐵))
44	43	orim1d 788	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (((℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → (∅ ∈ 𝐵 ∨ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})))
45	41, 44	mpd 13	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (∅ ∈ 𝐵 ∨ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
46	23, 45	jca 306	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ {∅} ∈ 𝐴) ∧ (∅ ∈ 𝐵 ∨ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})))
47		anddi 822	. . . . . 6 ⊢ ((((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∨ {∅} ∈ 𝐴) ∧ (∅ ∈ 𝐵 ∨ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) ↔ ((((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) ∨ (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))))
48	46, 47	sylib 122	. . . . 5 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ((((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) ∨ (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))))
49		simpl 109	. . . . . . 7 ⊢ (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) → {∅} ∈ 𝐴)
50		simpl 109	. . . . . . 7 ⊢ (({∅} ∈ 𝐴 ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → {∅} ∈ 𝐴)
51	49, 50	jaoi 717	. . . . . 6 ⊢ ((({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → {∅} ∈ 𝐴)
52	51	orim2i 762	. . . . 5 ⊢ (((((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) ∨ (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))) → ((((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) ∨ {∅} ∈ 𝐴))
53	48, 52	syl 14	. . . 4 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ((((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) ∨ {∅} ∈ 𝐴))
54	53	orcomd 730	. . 3 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ({∅} ∈ 𝐴 ∨ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))))
55		simpr 110	. . . . 5 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) → ∅ ∈ 𝐵)
56	55	orim1i 761	. . . 4 ⊢ ((((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → (∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})))
57	56	orim2i 762	. . 3 ⊢ (({∅} ∈ 𝐴 ∨ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))) → ({∅} ∈ 𝐴 ∨ (∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))))
58	54, 57	syl 14	. 2 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ({∅} ∈ 𝐴 ∨ (∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))))
59		3orass 983	. 2 ⊢ (({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) ↔ ({∅} ∈ 𝐴 ∨ (∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))))
60	58, 59	sylibr 134	1 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   ∨ w3o 979   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  ∃!wreu 2477  {crab 2479  Vcvv 2763  ∅c0 3450  {csn 3622  {cpr 3623  Oncon0 4398  ℩crio 5876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406  df-iota 5219  df-riota 5877

This theorem is referenced by:  acexmidlem1  5918