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Theorem ssdifsn 4351
Description: Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.)
Assertion
Ref Expression
ssdifsn (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵 ∧ ¬ 𝐶𝐴))

Proof of Theorem ssdifsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss3 3625 . . . 4 (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥𝐴 𝑥 ∈ (𝐵 ∖ {𝐶}))
2 eldifsn 4350 . . . . 5 (𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ (𝑥𝐵𝑥𝐶))
32ralbii 3009 . . . 4 (∀𝑥𝐴 𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝐶))
41, 3bitri 264 . . 3 (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝐶))
5 r19.26 3093 . . 3 (∀𝑥𝐴 (𝑥𝐵𝑥𝐶) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 𝑥𝐶))
64, 5bitri 264 . 2 (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 𝑥𝐶))
7 dfss3 3625 . . . 4 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
87bicomi 214 . . 3 (∀𝑥𝐴 𝑥𝐵𝐴𝐵)
9 neirr 2832 . . . . 5 ¬ 𝐶𝐶
10 neeq1 2885 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐶𝐶𝐶))
1110rspccv 3337 . . . . 5 (∀𝑥𝐴 𝑥𝐶 → (𝐶𝐴𝐶𝐶))
129, 11mtoi 190 . . . 4 (∀𝑥𝐴 𝑥𝐶 → ¬ 𝐶𝐴)
13 nelelne 2921 . . . . 5 𝐶𝐴 → (𝑥𝐴𝑥𝐶))
1413ralrimiv 2994 . . . 4 𝐶𝐴 → ∀𝑥𝐴 𝑥𝐶)
1512, 14impbii 199 . . 3 (∀𝑥𝐴 𝑥𝐶 ↔ ¬ 𝐶𝐴)
168, 15anbi12i 733 . 2 ((∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 𝑥𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐶𝐴))
176, 16bitri 264 1 (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵 ∧ ¬ 𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383  wcel 2030  wne 2823  wral 2941  cdif 3604  wss 3607  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-sn 4211
This theorem is referenced by:  logdivsqrle  30856  elsetrecslem  42770
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