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Theorem acexmidlemcase 5762
Description: Lemma for acexmid 5766. 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 (𝑣𝐴𝑢𝑦(𝐴𝑢𝑣𝑢)) and the choice function evaluated at 𝐵 is (𝑣𝐵𝑢𝑦(𝐵𝑢𝑣𝑢)). Other than the difference in notation these work just as (𝑦𝐴) and (𝑦𝐵) would if 𝑦 were a function as defined by df-fun 5120.

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 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})))
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
StepHypRef Expression
1 acexmidlem.a . . . . . . . . . . . . . 14 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
2 onsucelsucexmidlem 4439 . . . . . . . . . . . . . 14 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
31, 2eqeltri 2210 . . . . . . . . . . . . 13 𝐴 ∈ On
4 prid1g 3622 . . . . . . . . . . . . 13 (𝐴 ∈ On → 𝐴 ∈ {𝐴, 𝐵})
53, 4ax-mp 5 . . . . . . . . . . . 12 𝐴 ∈ {𝐴, 𝐵}
6 acexmidlem.c . . . . . . . . . . . 12 𝐶 = {𝐴, 𝐵}
75, 6eleqtrri 2213 . . . . . . . . . . 11 𝐴𝐶
8 eleq1 2200 . . . . . . . . . . . . . . 15 (𝑧 = 𝐴 → (𝑧𝑢𝐴𝑢))
98anbi1d 460 . . . . . . . . . . . . . 14 (𝑧 = 𝐴 → ((𝑧𝑢𝑣𝑢) ↔ (𝐴𝑢𝑣𝑢)))
109rexbidv 2436 . . . . . . . . . . . . 13 (𝑧 = 𝐴 → (∃𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑢𝑦 (𝐴𝑢𝑣𝑢)))
1110reueqd 2634 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)))
1211rspcv 2780 . . . . . . . . . . 11 (𝐴𝐶 → (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∃!𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)))
137, 12ax-mp 5 . . . . . . . . . 10 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∃!𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢))
14 riotacl 5737 . . . . . . . . . 10 (∃!𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢) → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ 𝐴)
1513, 14syl 14 . . . . . . . . 9 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ 𝐴)
16 elrabi 2832 . . . . . . . . . 10 ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ {∅, {∅}})
1716, 1eleq2s 2232 . . . . . . . . 9 ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ 𝐴 → (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ {∅, {∅}})
18 elpri 3545 . . . . . . . . 9 ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ {∅, {∅}} → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∨ (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = {∅}))
1915, 17, 183syl 17 . . . . . . . 8 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∨ (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = {∅}))
20 eleq1 2200 . . . . . . . . . 10 ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = {∅} → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) ∈ 𝐴 ↔ {∅} ∈ 𝐴))
2115, 20syl5ibcom 154 . . . . . . . . 9 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = {∅} → {∅} ∈ 𝐴))
2221orim2d 777 . . . . . . . 8 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∨ (𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = {∅}) → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∨ {∅} ∈ 𝐴)))
2319, 22mpd 13 . . . . . . 7 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∨ {∅} ∈ 𝐴))
24 acexmidlem.b . . . . . . . . . . . . . 14 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25 pp0ex 4108 . . . . . . . . . . . . . . 15 {∅, {∅}} ∈ V
2625rabex 4067 . . . . . . . . . . . . . 14 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ∈ V
2724, 26eqeltri 2210 . . . . . . . . . . . . 13 𝐵 ∈ V
2827prid2 3625 . . . . . . . . . . . 12 𝐵 ∈ {𝐴, 𝐵}
2928, 6eleqtrri 2213 . . . . . . . . . . 11 𝐵𝐶
30 eleq1 2200 . . . . . . . . . . . . . . 15 (𝑧 = 𝐵 → (𝑧𝑢𝐵𝑢))
3130anbi1d 460 . . . . . . . . . . . . . 14 (𝑧 = 𝐵 → ((𝑧𝑢𝑣𝑢) ↔ (𝐵𝑢𝑣𝑢)))
3231rexbidv 2436 . . . . . . . . . . . . 13 (𝑧 = 𝐵 → (∃𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑢𝑦 (𝐵𝑢𝑣𝑢)))
3332reueqd 2634 . . . . . . . . . . . 12 (𝑧 = 𝐵 → (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)))
3433rspcv 2780 . . . . . . . . . . 11 (𝐵𝐶 → (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∃!𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)))
3529, 34ax-mp 5 . . . . . . . . . 10 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∃!𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢))
36 riotacl 5737 . . . . . . . . . 10 (∃!𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ 𝐵)
3735, 36syl 14 . . . . . . . . 9 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ 𝐵)
38 elrabi 2832 . . . . . . . . . 10 ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ {∅, {∅}})
3938, 24eleq2s 2232 . . . . . . . . 9 ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ 𝐵 → (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ {∅, {∅}})
40 elpri 3545 . . . . . . . . 9 ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ {∅, {∅}} → ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅ ∨ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))
4137, 39, 403syl 17 . . . . . . . 8 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅ ∨ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))
42 eleq1 2200 . . . . . . . . . 10 ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅ → ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) ∈ 𝐵 ↔ ∅ ∈ 𝐵))
4337, 42syl5ibcom 154 . . . . . . . . 9 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅ → ∅ ∈ 𝐵))
4443orim1d 776 . . . . . . . 8 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (((𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = ∅ ∨ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → (∅ ∈ 𝐵 ∨ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})))
4541, 44mpd 13 . . . . . . 7 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (∅ ∈ 𝐵 ∨ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))
4623, 45jca 304 . . . . . 6 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∨ {∅} ∈ 𝐴) ∧ (∅ ∈ 𝐵 ∨ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})))
47 anddi 810 . . . . . 6 ((((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∨ {∅} ∈ 𝐴) ∧ (∅ ∈ 𝐵 ∨ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) ↔ ((((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) ∨ (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))))
4846, 47sylib 121 . . . . 5 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ((((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) ∨ (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))))
49 simpl 108 . . . . . . 7 (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) → {∅} ∈ 𝐴)
50 simpl 108 . . . . . . 7 (({∅} ∈ 𝐴 ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → {∅} ∈ 𝐴)
5149, 50jaoi 705 . . . . . 6 ((({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → {∅} ∈ 𝐴)
5251orim2i 750 . . . . 5 (((((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) ∨ (({∅} ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∨ ({∅} ∈ 𝐴 ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))) → ((((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) ∨ {∅} ∈ 𝐴))
5348, 52syl 14 . . . 4 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ((((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) ∨ {∅} ∈ 𝐴))
5453orcomd 718 . . 3 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ({∅} ∈ 𝐴 ∨ (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))))
55 simpr 109 . . . . 5 (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) → ∅ ∈ 𝐵)
5655orim1i 749 . . . 4 ((((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) → (∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})))
5756orim2i 750 . . 3 (({∅} ∈ 𝐴 ∨ (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ ∅ ∈ 𝐵) ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))) → ({∅} ∈ 𝐴 ∨ (∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))))
5854, 57syl 14 . 2 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ({∅} ∈ 𝐴 ∨ (∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))))
59 3orass 965 . 2 (({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})) ↔ ({∅} ∈ 𝐴 ∨ (∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}))))
6058, 59sylibr 133 1 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 697  w3o 961   = wceq 1331  wcel 1480  wral 2414  wrex 2415  ∃!wreu 2416  {crab 2418  Vcvv 2681  c0 3358  {csn 3522  {cpr 3523  Oncon0 4280  crio 5722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288  df-iota 5083  df-riota 5723
This theorem is referenced by:  acexmidlem1  5763
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