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Mathbox for Glauco Siliprandi |
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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 3454 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | ssrab2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | dfss3f 3987 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴) |
4 | rabidim1 3456 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | |
5 | 3, 4 | mprgbir 3066 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Ⅎwnfc 2888 {crab 3433 ⊆ wss 3963 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rab 3434 df-ss 3980 |
This theorem is referenced by: dmmptssf 45175 mptssid 45185 fnlimfvre 45630 limsupequzmpt2 45674 liminfequzmpt2 45747 pimltpnff 46659 pimgtmnff 46678 smflimlem2 46728 smflim 46733 smfpimcclem 46763 smfsupxr 46772 smfpimne2 46796 |
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