Theorem suceqsneq 38179

Description: One-to-one relationship between the successor operation and the singleton. (Contributed by Peter Mazsa, 31-Dec-2024.)

Assertion

Ref	Expression
suceqsneq	⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))

Proof of Theorem suceqsneq

Step	Hyp	Ref	Expression
1		suc11reg 9642	. 2 ⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)
2		sneqbg 4825	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	bitr4id 290	1 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  {csn 4608  suc csuc 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738  ax-reg 9615

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-eprel 5566  df-fr 5619  df-suc 6371

This theorem is referenced by: (None)