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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 | nfrab1 3122 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | ssrab2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | dfss3f 3595 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴) |
4 | rabidim1 3117 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | |
5 | 3, 4 | mprgbir 2927 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Ⅎwnfc 2751 {crab 2916 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-in 3581 df-ss 3588 |
This theorem is referenced by: dmmptssf 39438 mptssid 39450 fnlimfvre 39906 limsupequzmpt2 39950 liminfequzmpt2 40023 smflimlem2 40980 smflim 40985 smfpimcclem 41013 smfsupxr 41022 |
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