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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 8392 and efrirr 5019, 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 4835 | . . 3 ⊢ {𝑥} ∈ V | |
2 | eleq1 2676 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
3 | vsnid 4156 | . . . 4 ⊢ 𝑥 ∈ {𝑥} | |
4 | 2, 3 | spei 2249 | . . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥} |
5 | zfregcl 8382 | . . 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 4141 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
8 | ax9 1990 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) | |
9 | 8 | equcoms 1934 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦)) |
10 | 9 | com12 32 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦)) |
11 | 7, 10 | syl5bi 231 | . . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦)) |
12 | eleq1 2676 | . . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥})) | |
13 | 12 | notbid 307 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥})) |
14 | 13 | rspccv 3279 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥})) |
15 | 3, 14 | mt2i 131 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦) |
16 | 11, 15 | nsyli 154 | . . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥})) |
17 | 16 | con2d 128 | . . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})) |
18 | 17 | ralrimiv 2948 | . . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
19 | ralnex 2975 | . . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) | |
20 | 18, 19 | sylib 207 | . 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}) |
21 | 6, 20 | mt2 190 | 1 ⊢ ¬ 𝑥 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: elirr 8388 ruv 8390 dfac2 8836 nd1 9288 nd2 9289 nd3 9290 axunnd 9297 axregndlem1 9303 axregndlem2 9304 axregnd 9305 elpotr 30930 exnel 30952 distel 30953 |
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