Theorem pw1if 7548

Description: Expressing a truth value in terms of an if expression. (Contributed by Jim Kingdon, 10-Jan-2026.)

Assertion

Ref	Expression
pw1if	⊢ (𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)

Proof of Theorem pw1if

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → 𝑥 ∈ if(𝐴 = 1o, 1o, ∅))
2		elif 3638	. . . . . . 7 ⊢ (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ ((𝐴 = 1o ∧ 𝑥 ∈ 1o) ∨ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)))
3		noel 3516	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnan 937	. . . . . . . 8 ⊢ ¬ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)
5	4	biorfi 754	. . . . . . 7 ⊢ ((𝐴 = 1o ∧ 𝑥 ∈ 1o) ↔ ((𝐴 = 1o ∧ 𝑥 ∈ 1o) ∨ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)))
6	2, 5	bitr4i 187	. . . . . 6 ⊢ (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ (𝐴 = 1o ∧ 𝑥 ∈ 1o))
7	1, 6	sylib 122	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → (𝐴 = 1o ∧ 𝑥 ∈ 1o))
8	7	simprd 114	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → 𝑥 ∈ 1o)
9	7	simpld 112	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → 𝐴 = 1o)
10	8, 9	eleqtrrd 2314	. . 3 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → 𝑥 ∈ 𝐴)
11		elex2 2832	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
12		pw1m 7547	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ ∃𝑦 𝑦 ∈ 𝐴) → 𝐴 = 1o)
13	11, 12	sylan2 286	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 = 1o)
14		simpr 110	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
15	14, 13	eleqtrd 2313	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
16	13, 15, 6	sylanbrc 417	. . 3 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ if(𝐴 = 1o, 1o, ∅))
17	10, 16	impbida 600	. 2 ⊢ (𝐴 ∈ 𝒫 1o → (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ 𝑥 ∈ 𝐴))
18	17	eqrdv 2232	1 ⊢ (𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∅c0 3512  ifcif 3624  𝒫 cpw 3674  1_oc1o 6653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-suc 4497  df-1o 6660

This theorem is referenced by:  pw1map  16895