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}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))

Proof of Theorem ssdifsnOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss3 3741	. . . 4 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝐵 ∖ {𝐶}))
2		eldifsn 4453	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶))
3	2	ralbii 3129	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶))
4	1, 3	bitri 264	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶))
5		r19.26 3212	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶))
6	4, 5	bitri 264	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶))
7		dfss3 3741	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
8	7	bicomi 214	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ 𝐴 ⊆ 𝐵)
9		neirr 2952	. . . . 5 ⊢ ¬ 𝐶 ≠ 𝐶
10		neeq1 3005	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ≠ 𝐶 ↔ 𝐶 ≠ 𝐶))
11	10	rspccv 3457	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶 → (𝐶 ∈ 𝐴 → 𝐶 ≠ 𝐶))
12	9, 11	mtoi 190	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶 → ¬ 𝐶 ∈ 𝐴)
13		nelelne 3041	. . . . 5 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ≠ 𝐶))
14	13	ralrimiv 3114	. . . 4 ⊢ (¬ 𝐶 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶)
15	12, 14	impbii 199	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶 ↔ ¬ 𝐶 ∈ 𝐴)
16	8, 15	anbi12i 612	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))
17	6, 16	bitri 264	1 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))

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)