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Mirrors > Home > ILE Home > Th. List > exsnrex | GIF version |
Description: There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
Ref | Expression |
---|---|
exsnrex | ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2604 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 3425 | . . . . 5 ⊢ 𝑥 ∈ {𝑥} |
3 | eleq2 2142 | . . . . 5 ⊢ (𝑀 = {𝑥} → (𝑥 ∈ 𝑀 ↔ 𝑥 ∈ {𝑥})) | |
4 | 2, 3 | mpbiri 166 | . . . 4 ⊢ (𝑀 = {𝑥} → 𝑥 ∈ 𝑀) |
5 | 4 | pm4.71ri 384 | . . 3 ⊢ (𝑀 = {𝑥} ↔ (𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) |
6 | 5 | exbii 1536 | . 2 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) |
7 | df-rex 2354 | . 2 ⊢ (∃𝑥 ∈ 𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) | |
8 | 6, 7 | bitr4i 185 | 1 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∃wrex 2349 {csn 3398 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-sn 3404 |
This theorem is referenced by: (None) |
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