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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 | dfss2 3962 | . 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵) | |
2 | incom 4199 | . . . 4 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
3 | dfin5 3952 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | |
4 | 2, 3 | eqtri 2753 | . . 3 ⊢ (𝐵 ∩ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} |
5 | 4 | eqeq1i 2730 | . 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐵 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
6 | 1, 5 | bitri 274 | 1 ⊢ (𝐵 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 {crab 3418 ∩ cin 3943 ⊆ wss 3944 |
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-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-in 3951 df-ss 3961 |
This theorem is referenced by: qusker 33160 nsgqusf1olem3 33227 f1oresf1orab 46807 |
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