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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 2623 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 3479 | . . . . 5 ⊢ 𝑥 ∈ {𝑥} |
3 | eleq2 2152 | . . . . 5 ⊢ (𝑀 = {𝑥} → (𝑥 ∈ 𝑀 ↔ 𝑥 ∈ {𝑥})) | |
4 | 2, 3 | mpbiri 167 | . . . 4 ⊢ (𝑀 = {𝑥} → 𝑥 ∈ 𝑀) |
5 | 4 | pm4.71ri 385 | . . 3 ⊢ (𝑀 = {𝑥} ↔ (𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) |
6 | 5 | exbii 1542 | . 2 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) |
7 | df-rex 2366 | . 2 ⊢ (∃𝑥 ∈ 𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) | |
8 | 6, 7 | bitr4i 186 | 1 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1290 ∃wex 1427 ∈ wcel 1439 ∃wrex 2361 {csn 3450 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 df-v 2622 df-sn 3456 |
This theorem is referenced by: (None) |
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