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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 32153 | . 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴} | |
2 | vex 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | bj-sngltag 32164 | . . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴) |
5 | 4 | abbii 2726 | . 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴} |
6 | 1, 5 | eqtri 2632 | 1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 {csn 4125 sngl bj-csngl 32146 tag bj-ctag 32155 |
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-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-bj-sngl 32147 df-bj-tag 32156 |
This theorem is referenced by: bj-projval 32177 |
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