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Theorem nsspssun 3198
Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
nsspssun 𝐴𝐵𝐵 ⊊ (𝐴𝐵))

Proof of Theorem nsspssun
StepHypRef Expression
1 ssun2 3135 . . . 4 𝐵 ⊆ (𝐴𝐵)
21biantrur 291 . . 3 (¬ (𝐴𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
3 ssid 2992 . . . . 5 𝐵𝐵
43biantru 290 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐵𝐵))
5 unss 3145 . . . 4 ((𝐴𝐵𝐵𝐵) ↔ (𝐴𝐵) ⊆ 𝐵)
64, 5bitri 177 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) ⊆ 𝐵)
72, 6xchnxbir 616 . 2 𝐴𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
8 dfpss3 3058 . 2 (𝐵 ⊊ (𝐴𝐵) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
97, 8bitr4i 180 1 𝐴𝐵𝐵 ⊊ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wb 102  cun 2943  wss 2945  wpss 2946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pss 2961
This theorem is referenced by:  disjpss  3306
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