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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-taginv | Structured version Visualization version GIF version |
Description: Inverse of tagging. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-taginv | ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-snglinv 33835 | . 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴} | |
2 | bj-sngltag 33846 | . . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)) | |
3 | 2 | elv 3415 | . . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴) |
4 | 3 | abbii 2839 | . 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴} |
5 | 1, 4 | eqtri 2797 | 1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1508 ∈ wcel 2051 {cab 2753 Vcvv 3410 {csn 4436 sngl bj-csngl 33828 tag bj-ctag 33837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-rex 3089 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-sn 4437 df-pr 4439 df-bj-sngl 33829 df-bj-tag 33838 |
This theorem is referenced by: bj-projval 33859 |
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