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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-snglc | Structured version Visualization version GIF version |
Description: Characterization of the elements of 𝐴 in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-snglc | ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3141 | . 2 ⊢ (∃𝑥 ∈ 𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) | |
2 | bj-elsngl 34177 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 {𝐴} = {𝑥}) | |
3 | elisset 3503 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | 3 | pm4.71i 560 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) |
5 | 19.42v 1945 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) | |
6 | eleq1 2897 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
7 | 6 | eqcoms 2826 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
8 | 7 | pm5.32ri 576 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 8 | exbii 1839 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
10 | 4, 5, 9 | 3bitr2i 300 | . . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
11 | sneqbg 4766 | . . . . . . 7 ⊢ (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)) | |
12 | 11 | elv 3497 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
13 | eqcom 2825 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥}) | |
14 | 12, 13 | bitr3i 278 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ {𝐴} = {𝑥}) |
15 | 14 | anbi2i 622 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
16 | 15 | exbii 1839 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
17 | 10, 16 | bitri 276 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
18 | 1, 2, 17 | 3bitr4ri 305 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∃wrex 3136 Vcvv 3492 {csn 4557 sngl bj-csngl 34174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 df-dif 3936 df-un 3938 df-nul 4289 df-sn 4558 df-pr 4560 df-bj-sngl 34175 |
This theorem is referenced by: bj-snglinv 34181 bj-tagci 34193 bj-tagcg 34194 |
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