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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 33533 | . . 3 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) | |
2 | p0ex 5095 | . . . 4 ⊢ {∅} ∈ V | |
3 | 2 | biantru 525 | . . 3 ⊢ (sngl 𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V)) |
4 | 1, 3 | bitri 267 | . 2 ⊢ (𝐴 ∈ V ↔ (sngl 𝐴 ∈ V ∧ {∅} ∈ V)) |
5 | unexb 7235 | . 2 ⊢ ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V) | |
6 | df-bj-tag 33535 | . . . 4 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
7 | 6 | eqcomi 2786 | . . 3 ⊢ (sngl 𝐴 ∪ {∅}) = tag 𝐴 |
8 | 7 | eleq1i 2849 | . 2 ⊢ ((sngl 𝐴 ∪ {∅}) ∈ V ↔ tag 𝐴 ∈ V) |
9 | 4, 5, 8 | 3bitri 289 | 1 ⊢ (𝐴 ∈ V ↔ tag 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2106 Vcvv 3397 ∪ cun 3789 ∅c0 4140 {csn 4397 sngl bj-csngl 33525 tag bj-ctag 33534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-pw 4380 df-sn 4398 df-pr 4400 df-uni 4672 df-bj-sngl 33526 df-bj-tag 33535 |
This theorem is referenced by: bj-xtagex 33549 |
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