Theorem pw1if 7442

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 3617	. . . . . . 7 ⊢ (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ ((𝐴 = 1o ∧ 𝑥 ∈ 1o) ∨ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)))
3		noel 3498	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnan 936	. . . . . . . 8 ⊢ ¬ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)
5	4	biorfi 753	. . . . . . 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 2311	. . 3 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → 𝑥 ∈ 𝐴)
11		elex2 2819	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
12		pw1m 7441	. . . . 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 2310	. . . 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 2229	1 ⊢ (𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∅c0 3494  ifcif 3605  𝒫 cpw 3652  1_oc1o 6574

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-suc 4468  df-1o 6581

This theorem is referenced by:  pw1map  16596