Theorem sspw1or2 7446

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 2207	. . . . . 6 ⊢ (𝑠 = 𝑥 → (𝑗 ∈ 𝑠 ↔ 𝑗 ∈ 𝑥))
2	1	exbidv 1873	. . . . 5 ⊢ (𝑠 = 𝑥 → (∃𝑗 𝑗 ∈ 𝑠 ↔ ∃𝑗 𝑗 ∈ 𝑥))
3	2	elrab 2963	. . . 4 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥))
4	3	anbi1i 458	. . 3 ⊢ ((𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)) ↔ ((𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥) ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)))
5		en1m 7022	. . . . . 6 ⊢ (𝑥 ≈ 1o → ∃𝑗 𝑗 ∈ 𝑥)
6		en2m 7042	. . . . . 6 ⊢ (𝑥 ≈ 2o → ∃𝑗 𝑗 ∈ 𝑥)
7	5, 6	jaoi 724	. . . . 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 455	. . 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 2788	1 ⊢ {𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} = {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}

Syntax hints:   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {crab 2515  𝒫 cpw 3656   class class class wbr 4093  1_oc1o 6618  2_oc2o 6619   ≈ cen 6950

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-2o 6626  df-en 6953

This theorem is referenced by:  subupgr  16197