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Theorem disjpss 3272
Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
disjpss (((AB) = ∅ B ≠ ∅) → A ⊊ (AB))

Proof of Theorem disjpss
StepHypRef Expression
1 ssid 2958 . . . . . . . 8 BB
21biantru 286 . . . . . . 7 (BA ↔ (BA BB))
3 ssin 3153 . . . . . . 7 ((BA BB) ↔ B ⊆ (AB))
42, 3bitri 173 . . . . . 6 (BAB ⊆ (AB))
5 sseq2 2961 . . . . . 6 ((AB) = ∅ → (B ⊆ (AB) ↔ B ⊆ ∅))
64, 5syl5bb 181 . . . . 5 ((AB) = ∅ → (BAB ⊆ ∅))
7 ss0 3251 . . . . 5 (B ⊆ ∅ → B = ∅)
86, 7syl6bi 152 . . . 4 ((AB) = ∅ → (BAB = ∅))
98necon3ad 2241 . . 3 ((AB) = ∅ → (B ≠ ∅ → ¬ BA))
109imp 115 . 2 (((AB) = ∅ B ≠ ∅) → ¬ BA)
11 nsspssun 3164 . . 3 BAA ⊊ (BA))
12 uncom 3081 . . . 4 (BA) = (AB)
1312psseq2i 3028 . . 3 (A ⊊ (BA) ↔ A ⊊ (AB))
1411, 13bitri 173 . 2 BAA ⊊ (AB))
1510, 14sylib 127 1 (((AB) = ∅ B ≠ ∅) → A ⊊ (AB))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242  wne 2201  cun 2909  cin 2910  wss 2911  wpss 2912  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-bnd 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-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pss 2927  df-nul 3219
This theorem is referenced by: (None)
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