![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sselOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ssel 3973 as of 27-May-2024. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sselOLD | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3967 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | biimpi 215 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
3 | 2 | 19.21bi 2177 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | anim2d 610 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))) |
5 | 4 | eximdv 1912 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))) |
6 | dfclel 2806 | . 2 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴)) | |
7 | dfclel 2806 | . 2 ⊢ (𝐶 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)) | |
8 | 5, 6, 7 | 3imtr4g 295 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3473 df-in 3954 df-ss 3964 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |