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Theorem euen1 8193
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 8192 . 2 (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1𝑜)
2 reuv 3361 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
3 rabab 3363 . . 3 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
43breq1i 4811 . 2 ({𝑥 ∈ V ∣ 𝜑} ≈ 1𝑜 ↔ {𝑥𝜑} ≈ 1𝑜)
51, 2, 43bitr3i 290 1 (∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wb 196  ∃!weu 2607  {cab 2746  ∃!wreu 3052  {crab 3054  Vcvv 3340   class class class wbr 4804  1𝑜c1o 7723  cen 8120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1o 7730  df-en 8124
This theorem is referenced by:  euen1b  8194  modom  8328
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