Theorem mirimanoff 36628

Description: Mirimanoff's paradox: The class of grounded sets is not a set. Here, a set 𝑥 is grounded if there is no infinite membership-descending sequence of sets beginning with 𝑥. See Paragraph 1 of [Quine] p. 37 . (Contributed by Adrian Ducourtial, 11-Oct-2026.)

Hypothesis

Ref	Expression
mirimanoff.1	⊢ 𝑀 = {𝑥 ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}

Assertion

Ref	Expression
mirimanoff	⊢ 𝑀 ∉ V

Distinct variable group:   𝑓,𝑀,𝑛,𝑥

Proof of Theorem mirimanoff

Step	Hyp	Ref	Expression
1		mirimanoff.1	. . 3 ⊢ 𝑀 = {𝑥 ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}
2		rabab 3484	. . 3 ⊢ {𝑥 ∈ V ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))} = {𝑥 ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}
3	1, 2	eqtr4i 2789	. 2 ⊢ 𝑀 = {𝑥 ∈ V ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}
4	3	mirimanoffr 36627	1 ⊢ 𝑀 ∉ V

This proof depends on syntax axioms:  ¬ wn 3   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ∉ wnel 3065  ∀wral 3080  {crab 3416  Vcvv 3454  ∅c0 4281  suc csuc 6382   Fn wfn 6553  ‘cfv 6558  ωcom 7901

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-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-om 7902

This theorem is used by: (None)