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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 3385 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | ssrab2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | dfss3f 3959 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴) |
4 | rabidim1 3381 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | |
5 | 3, 4 | mprgbir 3153 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Ⅎwnfc 2961 {crab 3142 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-in 3943 df-ss 3952 |
This theorem is referenced by: dmmptssf 41494 mptssid 41503 fnlimfvre 41947 limsupequzmpt2 41991 liminfequzmpt2 42064 smflimlem2 43041 smflim 43046 smfpimcclem 43074 smfsupxr 43083 |
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