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Mirrors > Home > MPE Home > Th. List > exsnrex | Structured version Visualization version 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 | vsnid 4562 | . . . . 5 ⊢ 𝑥 ∈ {𝑥} | |
2 | eleq2 2878 | . . . . 5 ⊢ (𝑀 = {𝑥} → (𝑥 ∈ 𝑀 ↔ 𝑥 ∈ {𝑥})) | |
3 | 1, 2 | mpbiri 261 | . . . 4 ⊢ (𝑀 = {𝑥} → 𝑥 ∈ 𝑀) |
4 | 3 | pm4.71ri 564 | . . 3 ⊢ (𝑀 = {𝑥} ↔ (𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) |
5 | 4 | exbii 1849 | . 2 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) |
6 | df-rex 3112 | . 2 ⊢ (∃𝑥 ∈ 𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥})) | |
7 | 5, 6 | bitr4i 281 | 1 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃wrex 3107 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rex 3112 df-v 3443 df-sn 4526 |
This theorem is referenced by: frgrwopreg1 28103 frgrwopreg2 28104 |
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