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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 8862 and efrirr 5385, 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 5185 | . . 3 ⊢ {𝑥} ∈ V | |
2 | eleq1w 2843 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
3 | vsnid 4471 | . . . 4 ⊢ 𝑥 ∈ {𝑥} | |
4 | 2, 3 | speiv 1932 | . . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥} |
5 | zfregcl 8852 | . . 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 4452 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
8 | ax9 2064 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) | |
9 | 8 | equcoms 1978 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) |
10 | 9 | com12 32 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦)) |
11 | 7, 10 | syl5bi 234 | . . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦)) |
12 | eleq1w 2843 | . . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
13 | 12 | notbid 310 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥})) |
14 | 13 | rspccv 3527 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥})) |
15 | 3, 14 | mt2i 135 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦) |
16 | 11, 15 | nsyli 157 | . . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥})) |
17 | 16 | con2d 132 | . . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})) |
18 | 17 | ralrimiv 3126 | . . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
19 | ralnex 3178 | . . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) | |
20 | 18, 19 | sylib 210 | . 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
21 | 6, 20 | mt2 192 | 1 ⊢ ¬ 𝑥 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wex 1743 ∈ wcel 2051 ∀wral 3083 ∃wrex 3084 Vcvv 3410 {csn 4436 |
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-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: elirr 8855 ruv 8860 nd1 9806 nd2 9807 nd3 9808 axunnd 9815 axregndlem1 9821 axregndlem2 9822 axregnd 9823 elpotr 32579 exnel 32601 distel 32602 ralndv1 42740 |
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