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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 3806 | . 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵) | |
2 | incom 4028 | . . . 4 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
3 | dfin5 3800 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | |
4 | 2, 3 | eqtri 2802 | . . 3 ⊢ (𝐵 ∩ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} |
5 | 4 | eqeq1i 2783 | . 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐵 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
6 | 1, 5 | bitri 267 | 1 ⊢ (𝐵 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2107 {crab 3094 ∩ cin 3791 ⊆ wss 3792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-v 3400 df-in 3799 df-ss 3806 |
This theorem is referenced by: qusker 30411 f1oresf1orab 42340 |
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