Theorem snen1g 43486

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 2747	. . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2		vex 3492	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	sneqr 4865	. . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4		sneq 4658	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
5	3, 4	impbii 209	. . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
6	1, 5	bitri 275	. . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
7	6	exbii 1846	. 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8		en1 9086	. 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9		isset 3502	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	7, 8, 9	3bitr4i 303	1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V)

Syntax hints:   ↔ wb 206   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  {csn 4648   class class class wbr 5166  1_oc1o 8515   ≈ cen 9000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1o 8522  df-en 9004

This theorem is referenced by:  snen1el  43487