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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 2918 | . 2 ⊢ (∃𝑥 ∈ 𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) | |
2 | bj-elsngl 32956 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 {𝐴} = {𝑥}) | |
3 | elisset 3215 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | 3 | pm4.71i 664 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) |
5 | 19.42v 1918 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) | |
6 | eleq1 2689 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
7 | 6 | eqcoms 2630 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
8 | 7 | pm5.32ri 670 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 8 | exbii 1774 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
10 | 4, 5, 9 | 3bitr2i 288 | . . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
11 | vex 3203 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
12 | sneqbg 4374 | . . . . . . 7 ⊢ (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
14 | eqcom 2629 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥}) | |
15 | 13, 14 | bitr3i 266 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ {𝐴} = {𝑥}) |
16 | 15 | anbi2i 730 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
17 | 16 | exbii 1774 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
18 | 10, 17 | bitri 264 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
19 | 1, 2, 18 | 3bitr4ri 293 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 {csn 4177 sngl bj-csngl 32953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 df-bj-sngl 32954 |
This theorem is referenced by: bj-snglinv 32960 bj-tagci 32972 bj-tagcg 32973 |
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