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Theorem euen1 8265
Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
euen1 (∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1𝑜)

Proof of Theorem euen1
StepHypRef Expression
1 reuen1 8264 . 2 (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1𝑜)
2 reuv 3409 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
3 rabab 3411 . . 3 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
43breq1i 4850 . 2 ({𝑥 ∈ V ∣ 𝜑} ≈ 1𝑜 ↔ {𝑥𝜑} ≈ 1𝑜)
51, 2, 43bitr3i 293 1 (∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wb 198  ∃!weu 2608  {cab 2785  ∃!wreu 3091  {crab 3093  Vcvv 3385   class class class wbr 4843  1𝑜c1o 7792  cen 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-1o 7799  df-en 8196
This theorem is referenced by:  euen1b  8266  modom  8403
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