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Mirrors > Home > MPE Home > Th. List > axreg2 | Structured version Visualization version GIF version |
Description: Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.) |
Ref | Expression |
---|---|
axreg2 | ⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-reg 8380 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | |
2 | 1 | 19.23bi 2049 | 1 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: zfregcl 8382 zfregclOLD 8384 axregndlem2 9304 |
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