Theorem mirimanofff 36629

Description: Mirimanoff's paradox mirimanoff 36627 for founded sets. A set 𝑥 is founded if every set containing 𝑥 has a minimal element under membership. Compare with zfreg 9614 the defining condition of 𝑀 in the hypothesis. (Contributed by Adrian Ducourtial, 28-Oct-2026.)

Hypothesis

Ref	Expression
mirimanofff.1	⊢ 𝑀 = {𝑥 ∣ ∀𝑦(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝑦 (𝑧 ∩ 𝑦) = ∅)}

Assertion

Ref	Expression
mirimanofff	⊢ 𝑀 ∉ V

Distinct variable group:   𝑥,𝑀,𝑦,𝑧

Proof of Theorem mirimanofff

Step	Hyp	Ref	Expression
1		mirimanofff.1	. . 3 ⊢ 𝑀 = {𝑥 ∣ ∀𝑦(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝑦 (𝑧 ∩ 𝑦) = ∅)}
2		rabab 3484	. . 3 ⊢ {𝑥 ∈ V ∣ ∀𝑦(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝑦 (𝑧 ∩ 𝑦) = ∅)} = {𝑥 ∣ ∀𝑦(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝑦 (𝑧 ∩ 𝑦) = ∅)}
3	1, 2	eqtr4i 2789	. 2 ⊢ 𝑀 = {𝑥 ∈ V ∣ ∀𝑦(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝑦 (𝑧 ∩ 𝑦) = ∅)}
4	3	mirimanofffr 36628	1 ⊢ 𝑀 ∉ V

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570  {cab 2741   ∉ wnel 3065  ∃wrex 3090  {crab 3416  Vcvv 3454   ∩ cin 4092  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591

This theorem is used by: (None)