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Mirrors > Home > MPE Home > Th. List > dfss7 | Structured version Visualization version GIF version |
Description: Alternate definition of subclass relationship. (Contributed by AV, 1-Aug-2022.) |
Ref | Expression |
---|---|
dfss7 | ⊢ (𝐵 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3898 | . 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵) | |
2 | incom 4128 | . . . 4 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
3 | dfin5 3889 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | |
4 | 2, 3 | eqtri 2821 | . . 3 ⊢ (𝐵 ∩ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} |
5 | 4 | eqeq1i 2803 | . 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐵 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
6 | 1, 5 | bitri 278 | 1 ⊢ (𝐵 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 {crab 3110 ∩ cin 3880 ⊆ wss 3881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-rab 3115 df-in 3888 df-ss 3898 |
This theorem is referenced by: qusker 30969 f1oresf1orab 43845 |
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