![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2877 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
2 | 1 | sseq2i 4025 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
3 | ss2ab 4072 | . 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 1535 ∈ wcel 2106 {cab 2712 ⊆ 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-ss 3980 |
This theorem is referenced by: abssdvOLD 4079 rabss 4082 uniiunlem 4097 iunssf 5049 iunss 5050 abexd 5331 abex 5332 moabex 5470 reliun 5829 axdc2lem 10486 mptelee 28925 fpwrelmap 32751 hoidmvlelem1 46551 |
Copyright terms: Public domain | W3C validator |