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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssrab2f | Structured version Visualization version GIF version |
Description: Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ssrab2f.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
ssrab2f | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabidim1 38318 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | |
2 | 1 | rgen 2906 | . 2 ⊢ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴 |
3 | nfrab1 3099 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
4 | ssrab2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | 3, 4 | dfss3f 3560 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴) |
6 | 2, 5 | mpbir 220 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 {crab 2900 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-in 3547 df-ss 3554 |
This theorem is referenced by: fnlimfvre 38741 smflimlem2 39658 smflim 39663 |
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