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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 34178 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) | |
2 | bj-sngltag 34192 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) | |
3 | 1, 2 | syl5bb 284 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 {csn 4557 sngl bj-csngl 34174 tag bj-ctag 34183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-pr 4560 df-bj-sngl 34175 df-bj-tag 34184 |
This theorem is referenced by: (None) |
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