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Theorem euen1 8142
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 8141 . 2 (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1𝑜)
2 reuv 3325 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
3 rabab 3327 . . 3 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
43breq1i 4767 . 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 2571  {cab 2710  ∃!wreu 3016  {crab 3018  Vcvv 3304   class class class wbr 4760  1𝑜c1o 7673  cen 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-1o 7680  df-en 8073
This theorem is referenced by:  euen1b  8143  modom  8277
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