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Theorem uneqdifeqim 3302
 Description: Two ways that A and B can "partition" 𝐶 (when A and B don't overlap and A is a part of 𝐶). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
uneqdifeqim ((A𝐶 (AB) = ∅) → ((AB) = 𝐶 → (𝐶A) = B))

Proof of Theorem uneqdifeqim
StepHypRef Expression
1 uncom 3081 . . . 4 (BA) = (AB)
2 eqtr 2054 . . . . . 6 (((BA) = (AB) (AB) = 𝐶) → (BA) = 𝐶)
32eqcomd 2042 . . . . 5 (((BA) = (AB) (AB) = 𝐶) → 𝐶 = (BA))
4 difeq1 3049 . . . . . 6 (𝐶 = (BA) → (𝐶A) = ((BA) ∖ A))
5 difun2 3296 . . . . . 6 ((BA) ∖ A) = (BA)
6 eqtr 2054 . . . . . . 7 (((𝐶A) = ((BA) ∖ A) ((BA) ∖ A) = (BA)) → (𝐶A) = (BA))
7 incom 3123 . . . . . . . . . 10 (AB) = (BA)
87eqeq1i 2044 . . . . . . . . 9 ((AB) = ∅ ↔ (BA) = ∅)
9 disj3 3266 . . . . . . . . 9 ((BA) = ∅ ↔ B = (BA))
108, 9bitri 173 . . . . . . . 8 ((AB) = ∅ ↔ B = (BA))
11 eqtr 2054 . . . . . . . . . 10 (((𝐶A) = (BA) (BA) = B) → (𝐶A) = B)
1211expcom 109 . . . . . . . . 9 ((BA) = B → ((𝐶A) = (BA) → (𝐶A) = B))
1312eqcoms 2040 . . . . . . . 8 (B = (BA) → ((𝐶A) = (BA) → (𝐶A) = B))
1410, 13sylbi 114 . . . . . . 7 ((AB) = ∅ → ((𝐶A) = (BA) → (𝐶A) = B))
156, 14syl5com 26 . . . . . 6 (((𝐶A) = ((BA) ∖ A) ((BA) ∖ A) = (BA)) → ((AB) = ∅ → (𝐶A) = B))
164, 5, 15sylancl 392 . . . . 5 (𝐶 = (BA) → ((AB) = ∅ → (𝐶A) = B))
173, 16syl 14 . . . 4 (((BA) = (AB) (AB) = 𝐶) → ((AB) = ∅ → (𝐶A) = B))
181, 17mpan 400 . . 3 ((AB) = 𝐶 → ((AB) = ∅ → (𝐶A) = B))
1918com12 27 . 2 ((AB) = ∅ → ((AB) = 𝐶 → (𝐶A) = B))
2019adantl 262 1 ((A𝐶 (AB) = ∅) → ((AB) = 𝐶 → (𝐶A) = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∖ cdif 2908   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219 This theorem is referenced by: (None)
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