Theorem suceqsneq 39153

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 9584	. 2 ⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)
2		sneqbg 4808	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	bitr4id 293	1 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {csn 4589  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732  ax-reg 9550

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-eprel 5561  df-fr 5614  df-suc 6366

This theorem is referenced by: (None)