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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 3072 | . 2 ⊢ (∃𝑥 ∈ 𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) | |
2 | bj-elsngl 35849 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 {𝐴} = {𝑥}) | |
3 | elisset 2816 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | 3 | pm4.71i 561 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) |
5 | 19.42v 1958 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) | |
6 | eleq1 2822 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
7 | 6 | eqcoms 2741 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
8 | 7 | pm5.32ri 577 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 8 | exbii 1851 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
10 | 4, 5, 9 | 3bitr2i 299 | . . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
11 | sneqbg 4845 | . . . . . . 7 ⊢ (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)) | |
12 | 11 | elv 3481 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
13 | eqcom 2740 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥}) | |
14 | 12, 13 | bitr3i 277 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ {𝐴} = {𝑥}) |
15 | 14 | anbi2i 624 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
16 | 15 | exbii 1851 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
17 | 10, 16 | bitri 275 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
18 | 1, 2, 17 | 3bitr4ri 304 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3071 Vcvv 3475 {csn 4629 sngl bj-csngl 35846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3072 df-v 3477 df-un 3954 df-sn 4630 df-pr 4632 df-bj-sngl 35847 |
This theorem is referenced by: bj-snglinv 35853 bj-tagci 35865 bj-tagcg 35866 |
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