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Theorem disjpss 3255
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 2941 . . . . . . . 8 BB
21biantru 286 . . . . . . 7 (BA ↔ (BA BB))
3 ssin 3136 . . . . . . 7 ((BA BB) ↔ B ⊆ (AB))
42, 3bitri 173 . . . . . 6 (BAB ⊆ (AB))
5 sseq2 2944 . . . . . 6 ((AB) = ∅ → (B ⊆ (AB) ↔ B ⊆ ∅))
64, 5syl5bb 181 . . . . 5 ((AB) = ∅ → (BAB ⊆ ∅))
7 ss0 3234 . . . . 5 (B ⊆ ∅ → B = ∅)
86, 7syl6bi 152 . . . 4 ((AB) = ∅ → (BAB = ∅))
98necon3ad 2225 . . 3 ((AB) = ∅ → (B ≠ ∅ → ¬ BA))
109imp 115 . 2 (((AB) = ∅ B ≠ ∅) → ¬ BA)
11 nsspssun 3147 . . 3 BAA ⊊ (BA))
12 uncom 3064 . . . 4 (BA) = (AB)
1312psseq2i 3011 . . 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 1228  wne 2186  cun 2892  cin 2893  wss 2894  wpss 2895  c0 3201
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pss 2910  df-nul 3202
This theorem is referenced by: (None)
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