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Theorem unisn2 4714
Description: A version of unisn 4378 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
Assertion
Ref Expression
unisn2 {𝐴} ∈ {∅, 𝐴}

Proof of Theorem unisn2
StepHypRef Expression
1 unisng 4379 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4236 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2684 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4193 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 204 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4373 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4392 . . . 4 ∅ = ∅
8 0ex 4710 . . . . 5 ∅ ∈ V
98prid1 4237 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2680 . . 3 ∅ ∈ {∅, 𝐴}
116, 10syl6eqel 2692 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 174 1 {𝐴} ∈ {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1474  wcel 1976  Vcvv 3169  c0 3870  {csn 4121  {cpr 4123   cuni 4363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-nul 4709
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-sn 4122  df-pr 4124  df-uni 4364
This theorem is referenced by: (None)
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