|   | 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 2722 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
| 3 | 1, 2 | hbxfreq 2871 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 = wceq 1539 ∈ wcel 2107 {cab 2713 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: bnj1014 34976 bnj1145 35008 bnj1384 35047 bnj1398 35049 bnj1448 35062 bnj1450 35065 bnj1466 35068 bnj1463 35070 bnj1491 35072 bnj1497 35075 bnj1498 35076 bnj1520 35081 bnj1501 35082 | 
| Copyright terms: Public domain | W3C validator |