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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 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) | |
2 | bj-elsngl 32149 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 {𝐴} = {𝑥}) | |
3 | elisset 3188 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | 3 | pm4.71i 662 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) |
5 | 19.42v 1905 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) | |
6 | eleq1 2676 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
7 | 6 | eqcoms 2618 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
8 | 7 | pm5.32ri 668 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 8 | exbii 1764 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
10 | 4, 5, 9 | 3bitr2i 287 | . . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
11 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
12 | sneqbg 4314 | . . . . . . 7 ⊢ (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
14 | eqcom 2617 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥}) | |
15 | 13, 14 | bitr3i 265 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ {𝐴} = {𝑥}) |
16 | 15 | anbi2i 726 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
17 | 16 | exbii 1764 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
18 | 10, 17 | bitri 263 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
19 | 1, 2, 18 | 3bitr4ri 292 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 {csn 4125 sngl bj-csngl 32146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 df-bj-sngl 32147 |
This theorem is referenced by: bj-snglinv 32153 bj-tagci 32165 bj-tagcg 32166 |
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