Theorem dfprop2 38560

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

Assertion

Ref	Expression
dfprop2	⊢ PROP ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}

Distinct variable group:   𝑥,𝑛,𝑤,𝑧

Proof of Theorem dfprop2

Dummy variables 𝑎 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prop 38554	. . 3 ⊢ PROP = setrecs((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}))
2		dfprop1 38559	. . . . . . 7 ⊢ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ PROP
3		sstr2 3938	. . . . . . 7 ⊢ (𝑎 ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} → ({𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ PROP → 𝑎 ⊆ PROP))
4	2, 3	mpi 21	. . . . . 6 ⊢ (𝑎 ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} → 𝑎 ⊆ PROP)
5		rexeq 3315	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧) ↔ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)))
6	5	orbi2d 929	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → ((𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ↔ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧))))
7	6	rexeqbi1dv 3330	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ↔ ∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧))))
8	7	orbi1d 930	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → ((∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)) ↔ (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
9	8	abbidv 2826	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} = {𝑥 ∣ (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
10		eqid 2760	. . . . . . . . 9 ⊢ (𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}) = (𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
11		vex 3454	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
12	11	dfproplem 38555	. . . . . . . . 9 ⊢ {𝑥 ∣ (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ∈ V
13	9, 10, 12	fvmpt 6982	. . . . . . . 8 ⊢ (𝑎 ∈ V → ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘𝑎) = {𝑥 ∣ (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
14	13	elv 3455	. . . . . . 7 ⊢ ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘𝑎) = {𝑥 ∣ (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}
15		ssrexv 4001	. . . . . . . . . 10 ⊢ (𝑎 ⊆ PROP → (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) → ∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧))))
16	15	orim1d 981	. . . . . . . . 9 ⊢ (𝑎 ⊆ PROP → ((∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)) → (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
17		ssrexv 4001	. . . . . . . . . . . 12 ⊢ (𝑎 ⊆ PROP → (∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧) → ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)))
18	17	orim2d 982	. . . . . . . . . . 11 ⊢ (𝑎 ⊆ PROP → ((𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) → (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧))))
19	18	reximdv 3177	. . . . . . . . . 10 ⊢ (𝑎 ⊆ PROP → (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) → ∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧))))
20	19	orim1d 981	. . . . . . . . 9 ⊢ (𝑎 ⊆ PROP → ((∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)) → (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
21	16, 20	syld 48	. . . . . . . 8 ⊢ (𝑎 ⊆ PROP → ((∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)) → (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
22	21	ss2abdv 4013	. . . . . . 7 ⊢ (𝑎 ⊆ PROP → {𝑥 ∣ (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
23	14, 22	eqsstrid 3969	. . . . . 6 ⊢ (𝑎 ⊆ PROP → ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘𝑎) ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
24	4, 23	syl 18	. . . . 5 ⊢ (𝑎 ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} → ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘𝑎) ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
25	24	ax-gen 1828	. . . 4 ⊢ ∀𝑎(𝑎 ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} → ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘𝑎) ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
26	25	a1i 11	. . 3 ⊢ (⊤ → ∀𝑎(𝑎 ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} → ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘𝑎) ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}))
27	1, 26	setrec2v 9935	. 2 ⊢ (⊤ → PROP ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
28	27	mptru 1577	1 ⊢ PROP ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∀wal 1568   = wceq 1570  ⊤wtru 1571  {cab 2738  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899   ↦ cmpt 5186  ‘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:  dfprop  38561