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Mirrors > Home > MPE Home > Th. List > ssrabeq | Structured version Visualization version GIF version |
Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
Ref | Expression |
---|---|
ssrabeq | ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4059 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉 | |
2 | 1 | biantru 532 | . 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉)) |
3 | eqss 3985 | . 2 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉)) | |
4 | 2, 3 | bitr4i 280 | 1 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 {crab 3145 ⊆ wss 3939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-in 3946 df-ss 3955 |
This theorem is referenced by: difrab0eq 4422 |
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