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Theorem uneqsn 38818
Description: If a union of classes is equal to a singleton then at least one class is equal to the singleton while the other may be equal to the empty set. (Contributed by RP, 5-Jul-2021.)
Assertion
Ref Expression
uneqsn ((𝐴𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))

Proof of Theorem uneqsn
StepHypRef Expression
1 eqss 3813 . . . 4 ((𝐴𝐵) = {𝐶} ↔ ((𝐴𝐵) ⊆ {𝐶} ∧ {𝐶} ⊆ (𝐴𝐵)))
21a1i 11 . . 3 (𝐶 ∈ V → ((𝐴𝐵) = {𝐶} ↔ ((𝐴𝐵) ⊆ {𝐶} ∧ {𝐶} ⊆ (𝐴𝐵))))
3 unss 3986 . . . . . 6 ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ↔ (𝐴𝐵) ⊆ {𝐶})
43bicomi 215 . . . . 5 ((𝐴𝐵) ⊆ {𝐶} ↔ (𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}))
54a1i 11 . . . 4 (𝐶 ∈ V → ((𝐴𝐵) ⊆ {𝐶} ↔ (𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶})))
6 elun 3952 . . . . . 6 (𝐶 ∈ (𝐴𝐵) ↔ (𝐶𝐴𝐶𝐵))
7 snssg 4505 . . . . . . 7 (𝐶 ∈ V → (𝐶𝐴 ↔ {𝐶} ⊆ 𝐴))
8 snssg 4505 . . . . . . 7 (𝐶 ∈ V → (𝐶𝐵 ↔ {𝐶} ⊆ 𝐵))
97, 8orbi12d 933 . . . . . 6 (𝐶 ∈ V → ((𝐶𝐴𝐶𝐵) ↔ ({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵)))
106, 9syl5rbb 275 . . . . 5 (𝐶 ∈ V → (({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵) ↔ 𝐶 ∈ (𝐴𝐵)))
11 snssg 4505 . . . . 5 (𝐶 ∈ V → (𝐶 ∈ (𝐴𝐵) ↔ {𝐶} ⊆ (𝐴𝐵)))
1210, 11bitr2d 271 . . . 4 (𝐶 ∈ V → ({𝐶} ⊆ (𝐴𝐵) ↔ ({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵)))
135, 12anbi12d 618 . . 3 (𝐶 ∈ V → (((𝐴𝐵) ⊆ {𝐶} ∧ {𝐶} ⊆ (𝐴𝐵)) ↔ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵))))
14 or3or 38816 . . . . . 6 (({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵) ↔ (({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵) ∨ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵) ∨ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)))
1514anbi2i 611 . . . . 5 (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵)) ↔ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ (({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵) ∨ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵) ∨ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵))))
16 andi3or 38817 . . . . 5 (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ (({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵) ∨ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵) ∨ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵))) ↔ (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ∨ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵)) ∨ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵))))
1715, 16bitri 266 . . . 4 (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵)) ↔ (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ∨ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵)) ∨ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵))))
18 an4 638 . . . . . . 7 (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ↔ ((𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴) ∧ (𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵)))
19 eqss 3813 . . . . . . . . 9 (𝐴 = {𝐶} ↔ (𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴))
20 eqss 3813 . . . . . . . . 9 (𝐵 = {𝐶} ↔ (𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵))
2119, 20anbi12i 614 . . . . . . . 8 ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ↔ ((𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴) ∧ (𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵)))
2221bicomi 215 . . . . . . 7 (((𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴) ∧ (𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵)) ↔ (𝐴 = {𝐶} ∧ 𝐵 = {𝐶}))
2318, 22bitri 266 . . . . . 6 (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ↔ (𝐴 = {𝐶} ∧ 𝐵 = {𝐶}))
2423a1i 11 . . . . 5 (𝐶 ∈ V → (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ↔ (𝐴 = {𝐶} ∧ 𝐵 = {𝐶})))
25 an4 638 . . . . . 6 (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵)) ↔ ((𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴) ∧ (𝐵 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵)))
2619bicomi 215 . . . . . . . 8 ((𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴) ↔ 𝐴 = {𝐶})
2726a1i 11 . . . . . . 7 (𝐶 ∈ V → ((𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴) ↔ 𝐴 = {𝐶}))
28 sssn 4547 . . . . . . . . . 10 (𝐵 ⊆ {𝐶} ↔ (𝐵 = ∅ ∨ 𝐵 = {𝐶}))
2928a1i 11 . . . . . . . . 9 (𝐶 ∈ V → (𝐵 ⊆ {𝐶} ↔ (𝐵 = ∅ ∨ 𝐵 = {𝐶})))
3029anbi1d 617 . . . . . . . 8 (𝐶 ∈ V → ((𝐵 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵) ↔ ((𝐵 = ∅ ∨ 𝐵 = {𝐶}) ∧ ¬ {𝐶} ⊆ 𝐵)))
31 andir 1022 . . . . . . . . 9 (((𝐵 = ∅ ∨ 𝐵 = {𝐶}) ∧ ¬ {𝐶} ⊆ 𝐵) ↔ ((𝐵 = ∅ ∧ ¬ {𝐶} ⊆ 𝐵) ∨ (𝐵 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵)))
32 n0i 4121 . . . . . . . . . . . . 13 (𝐶𝐵 → ¬ 𝐵 = ∅)
338, 32syl6bir 245 . . . . . . . . . . . 12 (𝐶 ∈ V → ({𝐶} ⊆ 𝐵 → ¬ 𝐵 = ∅))
3433con2d 131 . . . . . . . . . . 11 (𝐶 ∈ V → (𝐵 = ∅ → ¬ {𝐶} ⊆ 𝐵))
3534pm4.71d 553 . . . . . . . . . 10 (𝐶 ∈ V → (𝐵 = ∅ ↔ (𝐵 = ∅ ∧ ¬ {𝐶} ⊆ 𝐵)))
36 eqimss2 3855 . . . . . . . . . . . 12 (𝐵 = {𝐶} → {𝐶} ⊆ 𝐵)
37 iman 390 . . . . . . . . . . . 12 ((𝐵 = {𝐶} → {𝐶} ⊆ 𝐵) ↔ ¬ (𝐵 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵))
3836, 37mpbi 221 . . . . . . . . . . 11 ¬ (𝐵 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵)
3938biorfi 953 . . . . . . . . . 10 ((𝐵 = ∅ ∧ ¬ {𝐶} ⊆ 𝐵) ↔ ((𝐵 = ∅ ∧ ¬ {𝐶} ⊆ 𝐵) ∨ (𝐵 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵)))
4035, 39syl6rbb 279 . . . . . . . . 9 (𝐶 ∈ V → (((𝐵 = ∅ ∧ ¬ {𝐶} ⊆ 𝐵) ∨ (𝐵 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵)) ↔ 𝐵 = ∅))
4131, 40syl5bb 274 . . . . . . . 8 (𝐶 ∈ V → (((𝐵 = ∅ ∨ 𝐵 = {𝐶}) ∧ ¬ {𝐶} ⊆ 𝐵) ↔ 𝐵 = ∅))
4230, 41bitrd 270 . . . . . . 7 (𝐶 ∈ V → ((𝐵 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵) ↔ 𝐵 = ∅))
4327, 42anbi12d 618 . . . . . 6 (𝐶 ∈ V → (((𝐴 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐴) ∧ (𝐵 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐵)) ↔ (𝐴 = {𝐶} ∧ 𝐵 = ∅)))
4425, 43syl5bb 274 . . . . 5 (𝐶 ∈ V → (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵)) ↔ (𝐴 = {𝐶} ∧ 𝐵 = ∅)))
45 an4 638 . . . . . 6 (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ↔ ((𝐴 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴) ∧ (𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵)))
46 sssn 4547 . . . . . . . . . 10 (𝐴 ⊆ {𝐶} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐶}))
4746a1i 11 . . . . . . . . 9 (𝐶 ∈ V → (𝐴 ⊆ {𝐶} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐶})))
4847anbi1d 617 . . . . . . . 8 (𝐶 ∈ V → ((𝐴 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐶}) ∧ ¬ {𝐶} ⊆ 𝐴)))
49 andir 1022 . . . . . . . . 9 (((𝐴 = ∅ ∨ 𝐴 = {𝐶}) ∧ ¬ {𝐶} ⊆ 𝐴) ↔ ((𝐴 = ∅ ∧ ¬ {𝐶} ⊆ 𝐴) ∨ (𝐴 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴)))
50 n0i 4121 . . . . . . . . . . . . 13 (𝐶𝐴 → ¬ 𝐴 = ∅)
517, 50syl6bir 245 . . . . . . . . . . . 12 (𝐶 ∈ V → ({𝐶} ⊆ 𝐴 → ¬ 𝐴 = ∅))
5251con2d 131 . . . . . . . . . . 11 (𝐶 ∈ V → (𝐴 = ∅ → ¬ {𝐶} ⊆ 𝐴))
5352pm4.71d 553 . . . . . . . . . 10 (𝐶 ∈ V → (𝐴 = ∅ ↔ (𝐴 = ∅ ∧ ¬ {𝐶} ⊆ 𝐴)))
54 eqimss2 3855 . . . . . . . . . . . 12 (𝐴 = {𝐶} → {𝐶} ⊆ 𝐴)
55 iman 390 . . . . . . . . . . . 12 ((𝐴 = {𝐶} → {𝐶} ⊆ 𝐴) ↔ ¬ (𝐴 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴))
5654, 55mpbi 221 . . . . . . . . . . 11 ¬ (𝐴 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴)
5756biorfi 953 . . . . . . . . . 10 ((𝐴 = ∅ ∧ ¬ {𝐶} ⊆ 𝐴) ↔ ((𝐴 = ∅ ∧ ¬ {𝐶} ⊆ 𝐴) ∨ (𝐴 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴)))
5853, 57syl6rbb 279 . . . . . . . . 9 (𝐶 ∈ V → (((𝐴 = ∅ ∧ ¬ {𝐶} ⊆ 𝐴) ∨ (𝐴 = {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴)) ↔ 𝐴 = ∅))
5949, 58syl5bb 274 . . . . . . . 8 (𝐶 ∈ V → (((𝐴 = ∅ ∨ 𝐴 = {𝐶}) ∧ ¬ {𝐶} ⊆ 𝐴) ↔ 𝐴 = ∅))
6048, 59bitrd 270 . . . . . . 7 (𝐶 ∈ V → ((𝐴 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴) ↔ 𝐴 = ∅))
6120bicomi 215 . . . . . . . 8 ((𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵) ↔ 𝐵 = {𝐶})
6261a1i 11 . . . . . . 7 (𝐶 ∈ V → ((𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵) ↔ 𝐵 = {𝐶}))
6360, 62anbi12d 618 . . . . . 6 (𝐶 ∈ V → (((𝐴 ⊆ {𝐶} ∧ ¬ {𝐶} ⊆ 𝐴) ∧ (𝐵 ⊆ {𝐶} ∧ {𝐶} ⊆ 𝐵)) ↔ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))
6445, 63syl5bb 274 . . . . 5 (𝐶 ∈ V → (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ↔ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))
6524, 44, 643orbi123d 1552 . . . 4 (𝐶 ∈ V → ((((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵)) ∨ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∧ ¬ {𝐶} ⊆ 𝐵)) ∨ ((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ (¬ {𝐶} ⊆ 𝐴 ∧ {𝐶} ⊆ 𝐵))) ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶}))))
6617, 65syl5bb 274 . . 3 (𝐶 ∈ V → (((𝐴 ⊆ {𝐶} ∧ 𝐵 ⊆ {𝐶}) ∧ ({𝐶} ⊆ 𝐴 ∨ {𝐶} ⊆ 𝐵)) ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶}))))
672, 13, 663bitrd 296 . 2 (𝐶 ∈ V → ((𝐴𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶}))))
68 snprc 4444 . . . . 5 𝐶 ∈ V ↔ {𝐶} = ∅)
6968biimpi 207 . . . 4 𝐶 ∈ V → {𝐶} = ∅)
7069eqeq2d 2816 . . 3 𝐶 ∈ V → ((𝐴𝐵) = {𝐶} ↔ (𝐴𝐵) = ∅))
7169eqeq2d 2816 . . . . . . . 8 𝐶 ∈ V → (𝐴 = {𝐶} ↔ 𝐴 = ∅))
7269eqeq2d 2816 . . . . . . . 8 𝐶 ∈ V → (𝐵 = {𝐶} ↔ 𝐵 = ∅))
7371, 72anbi12d 618 . . . . . . 7 𝐶 ∈ V → ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
7471anbi1d 617 . . . . . . 7 𝐶 ∈ V → ((𝐴 = {𝐶} ∧ 𝐵 = ∅) ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
7573, 74orbi12d 933 . . . . . 6 𝐶 ∈ V → (((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅)) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅))))
7672anbi2d 616 . . . . . 6 𝐶 ∈ V → ((𝐴 = ∅ ∧ 𝐵 = {𝐶}) ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
7775, 76orbi12d 933 . . . . 5 𝐶 ∈ V → ((((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅)) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅)) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅))))
78 pm4.25 920 . . . . . 6 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅)))
7978orbi1i 928 . . . . . 6 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅)) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅)) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅)))
8078, 79bitri 266 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅)) ∨ (𝐴 = ∅ ∧ 𝐵 = ∅)))
8177, 80syl6rbbr 281 . . . 4 𝐶 ∈ V → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅)) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶}))))
82 un00 4209 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
8382bicomi 215 . . . 4 ((𝐴𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))
84 df-3or 1101 . . . 4 (((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})) ↔ (((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅)) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))
8581, 83, 843bitr4g 305 . . 3 𝐶 ∈ V → ((𝐴𝐵) = ∅ ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶}))))
8670, 85bitrd 270 . 2 𝐶 ∈ V → ((𝐴𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶}))))
8767, 86pm2.61i 176 1 ((𝐴𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3o 1099   = wceq 1637  wcel 2156  Vcvv 3391  cun 3767  wss 3769  c0 4116  {csn 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-xor 1619  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-sn 4371
This theorem is referenced by:  clsk1indlem3  38838
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