Theorem pw1if 7406

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 3614	. . . . . . 7 ⊢ (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ ((𝐴 = 1o ∧ 𝑥 ∈ 1o) ∨ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)))
3		noel 3495	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnan 934	. . . . . . . 8 ⊢ ¬ (¬ 𝐴 = 1o ∧ 𝑥 ∈ ∅)
5	4	biorfi 751	. . . . . . 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 2309	. . 3 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ if(𝐴 = 1o, 1o, ∅)) → 𝑥 ∈ 𝐴)
11		elex2 2816	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
12		pw1m 7405	. . . . 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 2308	. . . 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 598	. 2 ⊢ (𝐴 ∈ 𝒫 1o → (𝑥 ∈ if(𝐴 = 1o, 1o, ∅) ↔ 𝑥 ∈ 𝐴))
18	17	eqrdv 2227	1 ⊢ (𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∅c0 3491  ifcif 3602  𝒫 cpw 3649  1_oc1o 6553

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-suc 4461  df-1o 6560

This theorem is referenced by:  pw1map  16320