Theorem snen1g 44717

Description: A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.)

Assertion

Ref	Expression
snen1g	⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V)

Proof of Theorem snen1g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2770	. . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	sneqr 4798	. . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4		sneq 4594	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
5	3, 4	impbii 212	. . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
6	1, 5	bitri 278	. . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
7	6	exbii 1881	. 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8		en1 9079	. 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9		isset 3468	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	7, 8, 9	3bitr4i 306	1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  {csn 4588   class class class wbr 5106  1_oc1o 8497   ≈ cen 8998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-1o 8504  df-en 9002

This theorem is used by:  snen1el  44718