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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 3456 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
| 2 | ssrab2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 1, 2 | dfss3f 3974 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴) | 
| 4 | rabidim1 3458 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | |
| 5 | 3, 4 | mprgbir 3067 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 Ⅎwnfc 2889 {crab 3435 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rab 3436 df-ss 3967 | 
| This theorem is referenced by: dmmptssf 45242 mptssid 45252 fnlimfvre 45694 limsupequzmpt2 45738 liminfequzmpt2 45811 pimltpnff 46723 pimgtmnff 46742 smflimlem2 46792 smflim 46797 smfpimcclem 46827 smfsupxr 46836 smfpimne2 46860 | 
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