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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 3144 | . 2 ⊢ (∃𝑥 ∈ 𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) | |
2 | bj-elsngl 34283 | . 2 ⊢ ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 {𝐴} = {𝑥}) | |
3 | elisset 3505 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | 3 | pm4.71i 562 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) |
5 | 19.42v 1954 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴)) | |
6 | eleq1 2900 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
7 | 6 | eqcoms 2829 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
8 | 7 | pm5.32ri 578 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 8 | exbii 1848 | . . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
10 | 4, 5, 9 | 3bitr2i 301 | . . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) |
11 | sneqbg 4774 | . . . . . . 7 ⊢ (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)) | |
12 | 11 | elv 3499 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
13 | eqcom 2828 | . . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥}) | |
14 | 12, 13 | bitr3i 279 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ {𝐴} = {𝑥}) |
15 | 14 | anbi2i 624 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
16 | 15 | exbii 1848 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
17 | 10, 16 | bitri 277 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥})) |
18 | 1, 2, 17 | 3bitr4ri 306 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 {csn 4567 sngl bj-csngl 34280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-dif 3939 df-un 3941 df-nul 4292 df-sn 4568 df-pr 4570 df-bj-sngl 34281 |
This theorem is referenced by: bj-snglinv 34287 bj-tagci 34299 bj-tagcg 34300 |
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