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| Mirrors > Home > MPE Home > Th. List > abss | Structured version Visualization version GIF version | ||
| Description: Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.) |
| Ref | Expression |
|---|---|
| abss | ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | abid2 2879 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
| 2 | 1 | sseq2i 4013 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
| 3 | ss2ab 4062 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) | |
| 4 | 2, 3 | bitr3i 277 | 1 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∈ wcel 2108 {cab 2714 ⊆ wss 3951 |
| 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-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ss 3968 |
| This theorem is referenced by: abssdvOLD 4069 rabss 4072 uniiunlem 4087 iunssf 5044 iunss 5045 abexd 5325 abex 5326 moabex 5464 reliun 5826 axdc2lem 10488 mptelee 28910 fpwrelmap 32744 hoidmvlelem1 46610 |
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