Theorem rabid1o 17034

Description: Converting between propositions and corresponding subsets of a singleton. (Contributed by Jim Kingdon, 31-Jul-2026.)

Assertion

Ref	Expression
rabid1o	⊢ ({𝑥 ∈ 1o ∣ 𝜑} = 1o ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem rabid1o

Step	Hyp	Ref	Expression
1		0lt1o 6713	. . 3 ⊢ ∅ ∈ 1o
2		elex2 2838	. . 3 ⊢ (∅ ∈ 1o → ∃𝑦 𝑦 ∈ 1o)
3		r19.3rmv 3618	. . 3 ⊢ (∃𝑦 𝑦 ∈ 1o → (𝜑 ↔ ∀𝑥 ∈ 1o 𝜑))
4	1, 2, 3	mp2b 8	. 2 ⊢ (𝜑 ↔ ∀𝑥 ∈ 1o 𝜑)
5		rabid2 2729	. 2 ⊢ (1o = {𝑥 ∈ 1o ∣ 𝜑} ↔ ∀𝑥 ∈ 1o 𝜑)
6		eqcom 2240	. 2 ⊢ (1o = {𝑥 ∈ 1o ∣ 𝜑} ↔ {𝑥 ∈ 1o ∣ 𝜑} = 1o)
7	4, 5, 6	3bitr2ri 209	1 ⊢ ({𝑥 ∈ 1o ∣ 𝜑} = 1o ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  ∅c0 3520  1_oc1o 6680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4259

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3715  df-suc 4516  df-1o 6687

This theorem is used by:  wexmiddc  17042  wexmiddifxylem  17045