Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elirrv | Structured version Visualization version GIF version |
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 9071 and efrirr 5539, 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 | snex 5335 | . . 3 ⊢ {𝑥} ∈ V | |
2 | eleq1w 2898 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
3 | vsnid 4605 | . . . 4 ⊢ 𝑥 ∈ {𝑥} | |
4 | 2, 3 | speivw 1976 | . . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥} |
5 | zfregcl 9061 | . . 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 4586 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
8 | ax9 2127 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) | |
9 | 8 | equcoms 2026 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) |
10 | 9 | com12 32 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦)) |
11 | 7, 10 | syl5bi 244 | . . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦)) |
12 | eleq1w 2898 | . . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
13 | 12 | notbid 320 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥})) |
14 | 13 | rspccv 3623 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥})) |
15 | 3, 14 | mt2i 139 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦) |
16 | 11, 15 | nsyli 160 | . . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥})) |
17 | 16 | con2d 136 | . . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})) |
18 | 17 | ralrimiv 3184 | . . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
19 | ralnex 3239 | . . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) | |
20 | 18, 19 | sylib 220 | . 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
21 | 6, 20 | mt2 202 | 1 ⊢ ¬ 𝑥 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wex 1779 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 Vcvv 3497 {csn 4570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-reg 9059 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-dif 3942 df-un 3944 df-nul 4295 df-sn 4571 df-pr 4573 |
This theorem is referenced by: elirr 9064 ruv 9069 nd1 10012 nd2 10013 nd3 10014 axunnd 10021 axregndlem1 10027 axregndlem2 10028 axregnd 10029 elpotr 33030 exnel 33051 distel 33052 ralndv1 43311 |
Copyright terms: Public domain | W3C validator |