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| Description: The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 9646 and efrirr 5664, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.) | 
| Ref | Expression | 
|---|---|
| elirrv | ⊢ ¬ 𝑥 ∈ 𝑥 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vsnex 5433 | . . 3 ⊢ {𝑥} ∈ V | |
| 2 | eleq1w 2823 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
| 3 | vsnid 4662 | . . . 4 ⊢ 𝑥 ∈ {𝑥} | |
| 4 | 2, 3 | speivw 1972 | . . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥} | 
| 5 | zfregcl 9635 | . . 3 ⊢ ({𝑥} ∈ V → (∃𝑦 𝑦 ∈ {𝑥} → ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})) | |
| 6 | 1, 4, 5 | mp2 9 | . 2 ⊢ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} | 
| 7 | velsn 4641 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
| 8 | ax9 2121 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) | |
| 9 | 8 | equcoms 2018 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) | 
| 10 | 9 | com12 32 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦)) | 
| 11 | 7, 10 | biimtrid 242 | . . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦)) | 
| 12 | eleq1w 2823 | . . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
| 13 | 12 | notbid 318 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥})) | 
| 14 | 13 | rspccv 3618 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥})) | 
| 15 | 3, 14 | mt2i 137 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦) | 
| 16 | 11, 15 | nsyli 157 | . . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥})) | 
| 17 | 16 | con2d 134 | . . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})) | 
| 18 | 17 | ralrimiv 3144 | . . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) | 
| 19 | ralnex 3071 | . . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) | |
| 20 | 18, 19 | sylib 218 | . 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) | 
| 21 | 6, 20 | mt2 200 | 1 ⊢ ¬ 𝑥 ∈ 𝑥 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∃wex 1778 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 Vcvv 3479 {csn 4625 | 
| 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 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-ral 3061 df-rex 3070 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: elirr 9638 nd1 10628 nd2 10629 nd3 10630 axunnd 10637 axregndlem1 10643 axregndlem2 10644 axregnd 10645 axnulg 35107 elpotr 35783 exnel 35804 distel 35805 ruvALT 42684 onsupmaxb 43256 ralndv1 47122 | 
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