Theorem dfprop 38838

Description: The set of sentences of propositional calculus is equal to the set of sentences that are either a variable encoded as a natural number, a negation of a sentence propositional calculus, or an implication between two sentences of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)

Assertion

Ref	Expression
dfprop	⊢ PROP = {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}

Distinct variable group:   𝑤,𝑛,𝑥,𝑧

Proof of Theorem dfprop

Step	Hyp	Ref	Expression
1		dfprop2 38837	. 2 ⊢ PROP ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}
2		dfprop1 38836	. 2 ⊢ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ PROP
3	1, 2	eqssi 3929	1 ⊢ PROP = {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570  {cab 2741  ∃wrex 3090  ‘cfv 6555  (class class class)co 7440  ℕcn 12359  propvar cpropvar 38824  prop¬cpropneg 38825  prop→cpropimp 38826  PROPcprop 38830

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-pow 5331  ax-pr 5395  ax-un 7772  ax-reg 9610  ax-inf2 9666  ax-cnex 11280  ax-1cn 11282  ax-addcl 11284

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-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-int 4911  df-iun 4956  df-iin 4957  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-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-r1 9792  df-rank 9793  df-scott 9953  df-setrecs 9989  df-nn 12360  df-prop 38831

This theorem is used by: (None)