Theorem dfprop1 38836

Description: The set of variables encoded as a natural number, negations of sentences of propositional calculus, and implications between sentences of propositional calculus is a subset of PROP. (Contributed by Thomas van Maaren, 21-Aug-2026.)

Assertion

Ref	Expression
dfprop1	⊢ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ PROP

Distinct variable groups:   𝑥,𝑛   𝑥,𝑤,𝑧

Proof of Theorem dfprop1

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . 6 ⊢ ((𝑧 ∈ PROP ∧ 𝑥 = (prop¬‘𝑧)) → 𝑥 = (prop¬‘𝑧))
2		negprop 38834	. . . . . . 7 ⊢ (𝑧 ∈ PROP → (prop¬‘𝑧) ∈ PROP)
3	2	adantr 486	. . . . . 6 ⊢ ((𝑧 ∈ PROP ∧ 𝑥 = (prop¬‘𝑧)) → (prop¬‘𝑧) ∈ PROP)
4	1, 3	eqeltrd 2864	. . . . 5 ⊢ ((𝑧 ∈ PROP ∧ 𝑥 = (prop¬‘𝑧)) → 𝑥 ∈ PROP)
5		df-rex 3091	. . . . . . . 8 ⊢ (∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧) ↔ ∃𝑤(𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧)))
6	5	anbi2i 635	. . . . . . 7 ⊢ ((𝑧 ∈ PROP ∧ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ↔ (𝑧 ∈ PROP ∧ ∃𝑤(𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))))
7		19.42v 1988	. . . . . . 7 ⊢ (∃𝑤(𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))) ↔ (𝑧 ∈ PROP ∧ ∃𝑤(𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))))
8	6, 7	bitr4i 281	. . . . . 6 ⊢ ((𝑧 ∈ PROP ∧ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ↔ ∃𝑤(𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))))
9		simprr 785	. . . . . . . 8 ⊢ ((𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))) → 𝑥 = (𝑤prop→𝑧))
10		simpl 488	. . . . . . . . 9 ⊢ ((𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧)) → 𝑤 ∈ PROP)
11		simpl 488	. . . . . . . . 9 ⊢ ((𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))) → 𝑧 ∈ PROP)
12		impprop 38835	. . . . . . . . 9 ⊢ ((𝑤 ∈ PROP ∧ 𝑧 ∈ PROP) → (𝑤prop→𝑧) ∈ PROP)
13	10, 11, 12	syl2an2 699	. . . . . . . 8 ⊢ ((𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))) → (𝑤prop→𝑧) ∈ PROP)
14	9, 13	eqeltrd 2864	. . . . . . 7 ⊢ ((𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))) → 𝑥 ∈ PROP)
15	14	exlimiv 1965	. . . . . 6 ⊢ (∃𝑤(𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))) → 𝑥 ∈ PROP)
16	8, 15	sylbi 220	. . . . 5 ⊢ ((𝑧 ∈ PROP ∧ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) → 𝑥 ∈ PROP)
17	4, 16	jaodan 972	. . . 4 ⊢ ((𝑧 ∈ PROP ∧ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧))) → 𝑥 ∈ PROP)
18	17	rexlimiva 3159	. . 3 ⊢ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) → 𝑥 ∈ PROP)
19		simpr 490	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 = (propvar ‘𝑛)) → 𝑥 = (propvar ‘𝑛))
20		varprop 38833	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (propvar ‘𝑛) ∈ PROP)
21	20	adantr 486	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 = (propvar ‘𝑛)) → (propvar ‘𝑛) ∈ PROP)
22	19, 21	eqeltrd 2864	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 = (propvar ‘𝑛)) → 𝑥 ∈ PROP)
23	22	rexlimiva 3159	. . 3 ⊢ (∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛) → 𝑥 ∈ PROP)
24	18, 23	jaoi 871	. 2 ⊢ ((∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)) → 𝑥 ∈ PROP)
25	24	abssi 3996	1 ⊢ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ PROP

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ⊆ wss 3900  ‘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:  dfprop2  38837  dfprop  38838