Theorem sspw1or2 7538

Description: The set of subsets of a given set with one or two elements can be expressed as elements of the power set or as inhabited elements of the power set. (Contributed by Jim Kingdon, 31-Mar-2026.)

Assertion

Ref	Expression
sspw1or2	⊢ {𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} = {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}

Distinct variable groups:   𝑉,𝑠   𝑗,𝑠,𝑥

Allowed substitution hints:   𝑉(𝑥, 𝑗)

Proof of Theorem sspw1or2

Step	Hyp	Ref	Expression
1		elequ2 2214	. . . . . 6 ⊢ (𝑠 = 𝑥 → (𝑗 ∈ 𝑠 ↔ 𝑗 ∈ 𝑥))
2	1	exbidv 1878	. . . . 5 ⊢ (𝑠 = 𝑥 → (∃𝑗 𝑗 ∈ 𝑠 ↔ ∃𝑗 𝑗 ∈ 𝑥))
3	2	elrab 2982	. . . 4 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥))
4	3	anbi1i 462	. . 3 ⊢ ((𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)) ↔ ((𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥) ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)))
5		en1m 7086	. . . . . 6 ⊢ (𝑥 ≈ 1o → ∃𝑗 𝑗 ∈ 𝑥)
6		en2m 7107	. . . . . 6 ⊢ (𝑥 ≈ 2o → ∃𝑗 𝑗 ∈ 𝑥)
7	5, 6	jaoi 728	. . . . 5 ⊢ ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) → ∃𝑗 𝑗 ∈ 𝑥)
8	7	biantrud 304	. . . 4 ⊢ ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) → (𝑥 ∈ 𝒫 𝑉 ↔ (𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥)))
9	8	pm5.32ri 459	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝑉 ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)) ↔ ((𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥) ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)))
10	4, 9	bitr4i 187	. 2 ⊢ ((𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)) ↔ (𝑥 ∈ 𝒫 𝑉 ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)))
11	10	rabbia2 2806	1 ⊢ {𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} = {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}

This proof depends on syntax axioms:   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {crab 2532  𝒫 cpw 3688   class class class wbr 4128  1_oc1o 6674  2_oc2o 6675   ≈ cen 7014

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-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1o 6681  df-2o 6682  df-en 7017

This theorem is used by:  subupgr  16497