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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 36361 | . . 3 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) | |
2 | p0ex 5375 | . . . 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 7732 | . 2 ⊢ ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V) | |
6 | df-bj-tag 36363 | . . . 4 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
7 | 6 | eqcomi 2735 | . . 3 ⊢ (sngl 𝐴 ∪ {∅}) = tag 𝐴 |
8 | 7 | eleq1i 2818 | . 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 205 ∧ wa 395 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 ∅c0 4317 {csn 4623 sngl bj-csngl 36353 tag bj-ctag 36362 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-pw 4599 df-sn 4624 df-pr 4626 df-uni 4903 df-bj-sngl 36354 df-bj-tag 36363 |
This theorem is referenced by: bj-xtagex 36377 |
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