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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-tagex | Structured version Visualization version GIF version | ||
| Description: A class is a set if and only if its tagging is a set. (Contributed by BJ, 6-Oct-2018.) | 
| Ref | Expression | 
|---|---|
| bj-tagex | ⊢ (𝐴 ∈ V ↔ tag 𝐴 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-snglex 36974 | . . 3 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) | |
| 2 | p0ex 5384 | . . . 4 ⊢ {∅} ∈ V | |
| 3 | 2 | biantru 529 | . . 3 ⊢ (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V)) | 
| 4 | 1, 3 | bitri 275 | . 2 ⊢ (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V)) | 
| 5 | unexb 7767 | . 2 ⊢ ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V) | |
| 6 | df-bj-tag 36976 | . . . 4 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
| 7 | 6 | eqcomi 2746 | . . 3 ⊢ (sngl 𝐴 ∪ {∅}) = tag 𝐴 | 
| 8 | 7 | eleq1i 2832 | . 2 ⊢ ((sngl 𝐴 ∪ {∅}) ∈ V ↔ tag 𝐴 ∈ V) | 
| 9 | 4, 5, 8 | 3bitri 297 | 1 ⊢ (𝐴 ∈ V ↔ tag 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅c0 4333 {csn 4626 sngl bj-csngl 36966 tag bj-ctag 36975 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-bj-sngl 36967 df-bj-tag 36976 | 
| This theorem is referenced by: bj-xtagex 36990 | 
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