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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-tagcg | Structured version Visualization version GIF version |
Description: Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-tagcg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-snglc 34802 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) | |
2 | bj-sngltag 34816 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) | |
3 | 1, 2 | syl5bb 286 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2114 {csn 4516 sngl bj-csngl 34798 tag bj-ctag 34807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rex 3059 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-sn 4517 df-pr 4519 df-bj-sngl 34799 df-bj-tag 34808 |
This theorem is referenced by: (None) |
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