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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 36582 | . 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴} | |
2 | bj-sngltag 36593 | . . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)) | |
3 | 2 | elv 3467 | . . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴) |
4 | 3 | abbii 2795 | . 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴} |
5 | 1, 4 | eqtri 2753 | 1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 {cab 2702 Vcvv 3461 {csn 4630 sngl bj-csngl 36575 tag bj-ctag 36584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rex 3060 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-sn 4631 df-pr 4633 df-bj-sngl 36576 df-bj-tag 36585 |
This theorem is referenced by: bj-projval 36606 |
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