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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 2724 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | hbxfreq 2869 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 = wceq 1539 ∈ wcel 2106 {cab 2715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 |
This theorem is referenced by: bnj1014 32941 bnj1145 32973 bnj1384 33012 bnj1398 33014 bnj1448 33027 bnj1450 33030 bnj1466 33033 bnj1463 33035 bnj1491 33037 bnj1497 33040 bnj1498 33041 bnj1520 33046 bnj1501 33047 |
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