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Theorem ssdifsnOLD 4455
Description: Obsolete proof of ssdifsn 4454 as of 31-May-2022. (Contributed by Emmett Weisz, 7-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ssdifsnOLD (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵 ∧ ¬ 𝐶𝐴))

Proof of Theorem ssdifsnOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss3 3741 . . . 4 (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥𝐴 𝑥 ∈ (𝐵 ∖ {𝐶}))
2 eldifsn 4453 . . . . 5 (𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ (𝑥𝐵𝑥𝐶))
32ralbii 3129 . . . 4 (∀𝑥𝐴 𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝐶))
41, 3bitri 264 . . 3 (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝐶))
5 r19.26 3212 . . 3 (∀𝑥𝐴 (𝑥𝐵𝑥𝐶) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 𝑥𝐶))
64, 5bitri 264 . 2 (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 𝑥𝐶))
7 dfss3 3741 . . . 4 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
87bicomi 214 . . 3 (∀𝑥𝐴 𝑥𝐵𝐴𝐵)
9 neirr 2952 . . . . 5 ¬ 𝐶𝐶
10 neeq1 3005 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐶𝐶𝐶))
1110rspccv 3457 . . . . 5 (∀𝑥𝐴 𝑥𝐶 → (𝐶𝐴𝐶𝐶))
129, 11mtoi 190 . . . 4 (∀𝑥𝐴 𝑥𝐶 → ¬ 𝐶𝐴)
13 nelelne 3041 . . . . 5 𝐶𝐴 → (𝑥𝐴𝑥𝐶))
1413ralrimiv 3114 . . . 4 𝐶𝐴 → ∀𝑥𝐴 𝑥𝐶)
1512, 14impbii 199 . . 3 (∀𝑥𝐴 𝑥𝐶 ↔ ¬ 𝐶𝐴)
168, 15anbi12i 612 . 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 382  wcel 2145  wne 2943  wral 3061  cdif 3720  wss 3723  {csn 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-sn 4317
This theorem is referenced by: (None)
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