Theorem dfprop1 38559

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 38557	. . . . . . 7 ⊢ (𝑧 ∈ PROP → (prop¬‘𝑧) ∈ PROP)
3	2	adantr 486	. . . . . 6 ⊢ ((𝑧 ∈ PROP ∧ 𝑥 = (prop¬‘𝑧)) → (prop¬‘𝑧) ∈ PROP)
4	1, 3	eqeltrd 2860	. . . . 5 ⊢ ((𝑧 ∈ PROP ∧ 𝑥 = (prop¬‘𝑧)) → 𝑥 ∈ PROP)
5		df-rex 3087	. . . . . . . 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 1986	. . . . . . 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 38558	. . . . . . . . 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 2860	. . . . . . 7 ⊢ ((𝑧 ∈ PROP ∧ (𝑤 ∈ PROP ∧ 𝑥 = (𝑤prop→𝑧))) → 𝑥 ∈ PROP)
15	14	exlimiv 1963	. . . . . 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 3155	. . 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 38556	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (propvar ‘𝑛) ∈ PROP)
21	20	adantr 486	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 = (propvar ‘𝑛)) → (propvar ‘𝑛) ∈ PROP)
22	19, 21	eqeltrd 2860	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 = (propvar ‘𝑛)) → 𝑥 ∈ PROP)
23	22	rexlimiva 3155	. . 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 4016	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 2145  {cab 2738  ∃wrex 3086   ⊆ wss 3899  ‘cfv 6528  (class class class)co 7409  ℕcn 12290  propvar cpropvar 38547  prop¬cpropneg 38548  prop→cpropimp 38549  PROPcprop 38553

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-reg 9564  ax-inf2 9620  ax-cnex 11213  ax-1cn 11215  ax-addcl 11217

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-om 7862  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-r1 9746  df-rank 9747  df-scott 9886  df-setrecs 9922  df-nn 12291  df-prop 38554

This theorem is used by:  dfprop2  38560  dfprop  38561