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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 8509 and efrirr 5095, 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 4908 | . . 3 ⊢ {𝑥} ∈ V | |
2 | eleq1 2689 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
3 | vsnid 4209 | . . . 4 ⊢ 𝑥 ∈ {𝑥} | |
4 | 2, 3 | spei 2261 | . . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥} |
5 | zfregcl 8499 | . . 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 4193 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
8 | ax9 2003 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) | |
9 | 8 | equcoms 1947 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) |
10 | 9 | com12 32 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦)) |
11 | 7, 10 | syl5bi 232 | . . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦)) |
12 | eleq1 2689 | . . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
13 | 12 | notbid 308 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥})) |
14 | 13 | rspccv 3306 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥})) |
15 | 3, 14 | mt2i 132 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦) |
16 | 11, 15 | nsyli 155 | . . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥})) |
17 | 16 | con2d 129 | . . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})) |
18 | 17 | ralrimiv 2965 | . . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
19 | ralnex 2992 | . . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) | |
20 | 18, 19 | sylib 208 | . 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
21 | 6, 20 | mt2 191 | 1 ⊢ ¬ 𝑥 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 |
This theorem is referenced by: elirr 8505 ruv 8507 dfac2 8953 nd1 9409 nd2 9410 nd3 9411 axunnd 9418 axregndlem1 9424 axregndlem2 9425 axregnd 9426 elpotr 31686 exnel 31708 distel 31709 |
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