Theorem dfprop1 38646

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 38644	. . . . . . 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 1985	. . . . . . 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 38645	. . . . . . . . 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 1962	. . . . . 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 38643	. . . . . 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 3993	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 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  ℕcn 12335  propvar cpropvar 38634  prop¬cpropneg 38635  prop→cpropimp 38636  PROPcprop 38640

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642  ax-cnex 11256  ax-1cn 11258  ax-addcl 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769  df-scott 9929  df-setrecs 9965  df-nn 12336  df-prop 38641

This theorem is used by:  dfprop2  38647  dfprop  38648