![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1317 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1317.1 | ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
bnj1317 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1317.1 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝜑} | |
2 | hbab1 2788 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | hbxfreq 2907 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1651 = wceq 1653 ∈ wcel 2157 {cab 2785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 |
This theorem is referenced by: bnj1014 31547 bnj1145 31578 bnj1384 31617 bnj1398 31619 bnj1448 31632 bnj1450 31635 bnj1466 31638 bnj1463 31640 bnj1491 31642 bnj1497 31645 bnj1498 31646 bnj1520 31651 bnj1501 31652 |
Copyright terms: Public domain | W3C validator |