Theorem pw1if 7574

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 3649	. . . . . . 7 ⊢ (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ ((𝐴 = 1o ∧ 𝑥 ∈ 1o) ∨ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)))
3		noel 3525	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnan 941	. . . . . . . 8 ⊢ ¬ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)
5	4	biorfi 758	. . . . . . 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 2318	. . 3 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → 𝑥 ∈ 𝐴)
11		elex2 2838	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
12		pw1m 7573	. . . . 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 2317	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
16	13, 15, 6	sylanbrc 421	. . 3 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ if(𝐴 = 1o, 1o, ∅))
17	10, 16	impbida 604	. 2 ⊢ (𝐴 ∈ 𝒫 1o → (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ 𝑥 ∈ 𝐴))
18	17	eqrdv 2236	1 ⊢ (𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∅c0 3520  ifcif 3635  𝒫 cpw 3685  1_oc1o 6670

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 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

This theorem 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-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-suc 4511  df-1o 6677

This theorem is referenced by:  pw1map  16939