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Mirrors > Home > MPE Home > Th. List > zfcndreg | Structured version Visualization version GIF version |
Description: Axiom of Regularity ax-reg 8850, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
zfcndreg | ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2088 | . 2 ⊢ Ⅎ𝑦∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) | |
2 | axregnd 9823 | . 2 ⊢ (𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) | |
3 | 1, 2 | exlimi 2148 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∀wal 1506 ∃wex 1743 ∈ wcel 2051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 ax-reg 8850 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ral 3088 df-rex 3089 df-v 3412 df-dif 3827 df-un 3829 df-nul 4174 df-sn 4437 df-pr 4439 |
This theorem is referenced by: (None) |
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