Theorem snen1g 44048

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 2763	. . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2		vex 3452	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	sneqr 4792	. . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4		sneq 4586	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
5	3, 4	impbii 211	. . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
6	1, 5	bitri 277	. . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
7	6	exbii 1862	. 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8		en1 8994	. 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9		isset 3462	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	7, 8, 9	3bitr4i 305	1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V)

Syntax hints:   ↔ wb 208   = wceq 1554  ∃wex 1793   ∈ wcel 2136  Vcvv 3448  {csn 4576   class class class wbr 5094  1_oc1o 8418   ≈ cen 8913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-1o 8425  df-en 8917

This theorem is referenced by:  snen1el  44049