Theorem dfprop2 38837

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 38831	. . 3 ⊢ PROP = setrecs((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}))
2		dfprop1 38836	. . . . . . 7 ⊢ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ PROP
3		sstr2 3920	. . . . . . 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 3319	. . . . . . . . . . . . 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 3334	. . . . . . . . . . 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 2829	. . . . . . . . 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 2763	. . . . . . . . 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 3458	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
12	11	dfproplem 38832	. . . . . . . . 9 ⊢ {𝑥 ∣ (∃𝑧 ∈ 𝑎 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑎 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ∈ V
13	9, 10, 12	fvmpt 7013	. . . . . . . 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 3459	. . . . . . 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 3983	. . . . . . . . . 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 3983	. . . . . . . . . . . 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 3181	. . . . . . . . . 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 3993	. . . . . . 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 3951	. . . . . 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 10002	. 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 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ↦ cmpt 5189  ‘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:  dfprop  38838