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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 36956 | . . 3 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) | |
2 | p0ex 5390 | . . . 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 7766 | . 2 ⊢ ((sngl 𝐴 ∈ V ∧ {∅} ∈ V) ↔ (sngl 𝐴 ∪ {∅}) ∈ V) | |
6 | df-bj-tag 36958 | . . . 4 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
7 | 6 | eqcomi 2744 | . . 3 ⊢ (sngl 𝐴 ∪ {∅}) = tag 𝐴 |
8 | 7 | eleq1i 2830 | . 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 2106 Vcvv 3478 ∪ cun 3961 ∅c0 4339 {csn 4631 sngl bj-csngl 36948 tag bj-ctag 36957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-bj-sngl 36949 df-bj-tag 36958 |
This theorem is referenced by: bj-xtagex 36972 |
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