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| Description: Axiom of Regularity ax-reg 9633, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| zfcndreg | ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfe1 2149 | . 2 ⊢ Ⅎ𝑦∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) | |
| 2 | axregnd 10645 | . 2 ⊢ (𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) | |
| 3 | 1, 2 | exlimi 2216 | 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 395 ∀wal 1537 ∃wex 1778 ∈ wcel 2107 | 
| 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-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 ax-sep 5295 ax-pr 5431 ax-reg 9633 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: (None) | 
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