Theorem sselOLD 3911

Description: Obsolete version of ssel 3910 as of 27-May-2024. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sselOLD	⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))

Proof of Theorem sselOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3903	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	biimpi 215	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3	2	19.21bi 2184	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4	3	anim2d 611	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)))
5	4	eximdv 1921	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)))
6		dfclel 2818	. 2 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴))
7		dfclel 2818	. 2 ⊢ (𝐶 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))
8	5, 6, 7	3imtr4g 295	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by: (None)