Theorem varprop 38643

Description: Variables encoded as natural numbers are sentences of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)

Assertion

Ref	Expression
varprop	⊢ (𝑛 ∈ ℕ → (propvar ‘𝑛) ∈ PROP)

Proof of Theorem varprop

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prop 38641	. . 3 ⊢ PROP = setrecs((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}))
2		0ex 5260	. . . 4 ⊢ ∅ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝑛 ∈ ℕ → ∅ ∈ V)
4		0ss 4349	. . . 4 ⊢ ∅ ⊆ PROP
5	4	a1i 11	. . 3 ⊢ (𝑛 ∈ ℕ → ∅ ⊆ PROP)
6	1, 3, 5	setrec1 9972	. 2 ⊢ (𝑛 ∈ ℕ → ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘∅) ⊆ PROP)
7		rspe 3252	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 = (propvar ‘𝑛)) → ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))
8	7	olcd 888	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 = (propvar ‘𝑛)) → (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)))
9	8	ex 418	. . . . 5 ⊢ (𝑛 ∈ ℕ → (𝑥 = (propvar ‘𝑛) → (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
10	9	alrimiv 1959	. . . 4 ⊢ (𝑛 ∈ ℕ → ∀𝑥(𝑥 = (propvar ‘𝑛) → (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
11		fvex 6898	. . . . 5 ⊢ (propvar ‘𝑛) ∈ V
12		elab6g 3622	. . . . 5 ⊢ ((propvar ‘𝑛) ∈ V → ((propvar ‘𝑛) ∈ {𝑥 ∣ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ↔ ∀𝑥(𝑥 = (propvar ‘𝑛) → (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)))))
13	11, 12	ax-mp 5	. . . 4 ⊢ ((propvar ‘𝑛) ∈ {𝑥 ∣ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ↔ ∀𝑥(𝑥 = (propvar ‘𝑛) → (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
14	10, 13	sylibr 237	. . 3 ⊢ (𝑛 ∈ ℕ → (propvar ‘𝑛) ∈ {𝑥 ∣ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
15		rexeq 3315	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧) ↔ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)))
16	15	orbi2d 929	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ↔ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧))))
17	16	rexeqbi1dv 3330	. . . . . . 7 ⊢ (𝑦 = ∅ → (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ↔ ∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧))))
18	17	orbi1d 930	. . . . . 6 ⊢ (𝑦 = ∅ → ((∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)) ↔ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))))
19	18	abbidv 2826	. . . . 5 ⊢ (𝑦 = ∅ → {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} = {𝑥 ∣ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
20		eqid 2760	. . . . 5 ⊢ (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))})
21	2	dfproplem 38642	. . . . 5 ⊢ {𝑥 ∣ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ∈ V
22	19, 20, 21	fvmpt 6993	. . . 4 ⊢ (∅ ∈ V → ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘∅) = {𝑥 ∣ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})
23	2, 22	ax-mp 5	. . 3 ⊢ ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘∅) = {𝑥 ∣ (∃𝑧 ∈ ∅ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ ∅ 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}
24	14, 23	eleqtrrdi 2871	. 2 ⊢ (𝑛 ∈ ℕ → (propvar ‘𝑛) ∈ ((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))})‘∅))
25	6, 24	sseldd 3909	1 ⊢ (𝑛 ∈ ℕ → (propvar ‘𝑛) ∈ PROP)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∅c0 4278   ↦ cmpt 5185  ‘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:  dfprop1  38646