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Mirrors > Home > MPE Home > Th. List > sselOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ssel 3881 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 3874 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | biimpi 219 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
3 | 2 | 19.21bi 2187 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | anim2d 615 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))) |
5 | 4 | eximdv 1919 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))) |
6 | dfclel 2832 | . 2 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴)) | |
7 | dfclel 2832 | . 2 ⊢ (𝐶 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)) | |
8 | 5, 6, 7 | 3imtr4g 300 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∀wal 1537 = wceq 1539 ∃wex 1782 ∈ wcel 2112 ⊆ wss 3854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-v 3409 df-in 3861 df-ss 3871 |
This theorem is referenced by: (None) |
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