Theorem snen1g 41029

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 2745	. . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	sneqr 4768	. . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4		sneq 4568	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
5	3, 4	impbii 208	. . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
6	1, 5	bitri 274	. . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
7	6	exbii 1851	. 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8		en1 8765	. 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9		isset 3435	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	7, 8, 9	3bitr4i 302	1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V)

Syntax hints:   ↔ wb 205   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422  {csn 4558   class class class wbr 5070  1_oc1o 8260   ≈ cen 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1o 8267  df-en 8692

This theorem is referenced by:  snen1el  41030