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Theorem suceqsneq 37616
Description: One-to-one relationship between the successor operation and the singleton. (Contributed by Peter Mazsa, 31-Dec-2024.)
Assertion
Ref Expression
suceqsneq (𝐴𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))

Proof of Theorem suceqsneq
StepHypRef Expression
1 suc11reg 9616 . 2 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
2 sneqbg 4839 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
31, 2bitr4id 290 1 (𝐴𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {csn 4623  suc csuc 6360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722  ax-reg 9589
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-eprel 5573  df-fr 5624  df-suc 6364
This theorem is referenced by: (None)
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