![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 36493 | . . 3 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) | |
2 | p0ex 5388 | . . . 4 ⊢ {∅} ∈ V | |
3 | 2 | biantru 528 | . . 3 ⊢ (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V)) |
4 | 1, 3 | bitri 274 | . 2 ⊢ (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V)) |
5 | unexb 7758 | . 2 ⊢ ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V) | |
6 | df-bj-tag 36495 | . . . 4 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
7 | 6 | eqcomi 2737 | . . 3 ⊢ (sngl 𝐴 ∪ {∅}) = tag 𝐴 |
8 | 7 | eleq1i 2820 | . 2 ⊢ ((sngl 𝐴 ∪ {∅}) ∈ V ↔ tag 𝐴 ∈ V) |
9 | 4, 5, 8 | 3bitri 296 | 1 ⊢ (𝐴 ∈ V ↔ tag 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2098 Vcvv 3473 ∪ cun 3947 ∅c0 4326 {csn 4632 sngl bj-csngl 36485 tag bj-ctag 36494 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-pw 4608 df-sn 4633 df-pr 4635 df-uni 4913 df-bj-sngl 36486 df-bj-tag 36495 |
This theorem is referenced by: bj-xtagex 36509 |
Copyright terms: Public domain | W3C validator |