| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 3969 | . 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵) | |
| 2 | incom 4209 | . . . 4 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
| 3 | dfin5 3959 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | |
| 4 | 2, 3 | eqtri 2765 | . . 3 ⊢ (𝐵 ∩ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} |
| 5 | 4 | eqeq1i 2742 | . 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐵 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
| 6 | 1, 5 | bitri 275 | 1 ⊢ (𝐵 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {crab 3436 ∩ cin 3950 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-in 3958 df-ss 3968 |
| This theorem is referenced by: qusker 33377 nsgqusf1olem3 33443 f1oresf1orab 47301 |
| Copyright terms: Public domain | W3C validator |