Theorem moabex 5470

Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)

Assertion

Ref	Expression
moabex	⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Proof of Theorem moabex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mo 2538	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		abss 4073	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 ∈ {𝑦}))
3		velsn 4647	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
4	3	imbi2i 336	. . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ {𝑦}) ↔ (𝜑 → 𝑥 = 𝑦))
5	4	albii 1816	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))
6	2, 5	bitri 275	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))
7		vsnex 5440	. . . . 5 ⊢ {𝑦} ∈ V
8	7	ssex 5327	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
9	6, 8	sylbir 235	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
10	9	exlimiv 1928	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
11	1, 10	sylbi 217	1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1776   ∈ wcel 2106  ∃*wmo 2536  {cab 2712  Vcvv 3478   ⊆ wss 3963  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-sn 4632  df-pr 4634

This theorem is referenced by:  rmorabex  5471  euabex  5472  satfv0  35343