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Theorem bj-1uplex 33300
 Description: A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-1uplex (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)

Proof of Theorem bj-1uplex
StepHypRef Expression
1 bj-pr11val 33297 . . 3 pr1𝐴⦆ = 𝐴
2 bj-pr1ex 33298 . . 3 (⦅𝐴⦆ ∈ V → pr1𝐴⦆ ∈ V)
31, 2syl5eqelr 2842 . 2 (⦅𝐴⦆ ∈ V → 𝐴 ∈ V)
4 df-bj-1upl 33290 . . 3 𝐴⦆ = ({∅} × tag 𝐴)
5 p0ex 5000 . . . 4 {∅} ∈ V
6 bj-xtagex 33281 . . . 4 ({∅} ∈ V → (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V))
75, 6ax-mp 5 . . 3 (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V)
84, 7syl5eqel 2841 . 2 (𝐴 ∈ V → ⦅𝐴⦆ ∈ V)
93, 8impbii 199 1 (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2137  Vcvv 3338  ∅c0 4056  {csn 4319   × cxp 5262  tag bj-ctag 33266  ⦅bj-c1upl 33289  pr1 bj-cpr1 33292 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-xp 5270  df-rel 5271  df-cnv 5272  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-bj-sngl 33258  df-bj-tag 33267  df-bj-proj 33283  df-bj-1upl 33290  df-bj-pr1 33293 This theorem is referenced by:  bj-2uplex  33314
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