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| Mirrors > Home > MPE Home > Th. List > rabsssn | Structured version Visualization version GIF version | ||
| Description: Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| rabsssn | ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rab 3437 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} | |
| 2 | df-sn 4627 | . . 3 ⊢ {𝑋} = {𝑥 ∣ 𝑥 = 𝑋} | |
| 3 | 1, 2 | sseq12i 4014 | . 2 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 = 𝑋}) | 
| 4 | ss2ab 4062 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 = 𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋)) | |
| 5 | impexp 450 | . . . 4 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋) ↔ (𝑥 ∈ 𝑉 → (𝜑 → 𝑥 = 𝑋))) | |
| 6 | 5 | albii 1819 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝑉 → (𝜑 → 𝑥 = 𝑋))) | 
| 7 | df-ral 3062 | . . 3 ⊢ (∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝑉 → (𝜑 → 𝑥 = 𝑋))) | |
| 8 | 6, 7 | bitr4i 278 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋)) | 
| 9 | 3, 4, 8 | 3bitri 297 | 1 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 {crab 3436 ⊆ wss 3951 {csn 4626 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-ss 3968 df-sn 4627 | 
| This theorem is referenced by: constrfin 33787 suppmptcfin 48292 linc1 48342 | 
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