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Theorem disjpss 3278
 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 2964 . . . . . . . 8 𝐵𝐵
21biantru 286 . . . . . . 7 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐵))
3 ssin 3159 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
42, 3bitri 173 . . . . . 6 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
5 sseq2 2967 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
64, 5syl5bb 181 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 ⊆ ∅))
7 ss0 3257 . . . . 5 (𝐵 ⊆ ∅ → 𝐵 = ∅)
86, 7syl6bi 152 . . . 4 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 = ∅))
98necon3ad 2247 . . 3 ((𝐴𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵𝐴))
109imp 115 . 2 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵𝐴)
11 nsspssun 3170 . . 3 𝐵𝐴𝐴 ⊊ (𝐵𝐴))
12 uncom 3087 . . . 4 (𝐵𝐴) = (𝐴𝐵)
1312psseq2i 3034 . . 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 1243   ≠ wne 2204   ∪ cun 2915   ∩ cin 2916   ⊆ wss 2917   ⊊ wpss 2918  ∅c0 3224 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pss 2933  df-nul 3225 This theorem is referenced by: (None)
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