Theorem trpred0 33077

Description: The class of transitive predecessors is empty when 𝐴 is empty. (Contributed by Scott Fenton, 30-Apr-2012.)

Assertion

Ref	Expression
trpred0	⊢ TrPred(𝑅, ∅, 𝑋) = ∅

Proof of Theorem trpred0

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

Step	Hyp	Ref	Expression
1		dftrpred2 33060	. 2 ⊢ TrPred(𝑅, ∅, 𝑋) = ∪ 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖)
2		pred0 6180	. . . . . . . . . . 11 ⊢ Pred(𝑅, ∅, 𝑦) = ∅
3	2	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑎 → Pred(𝑅, ∅, 𝑦) = ∅)
4	3	iuneq2i 4942	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦) = ∪ 𝑦 ∈ 𝑎 ∅
5		iun0 4987	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ 𝑎 ∅ = ∅
6	4, 5	eqtri 2846	. . . . . . . 8 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦) = ∅
7	6	mpteq2i 5160	. . . . . . 7 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)) = (𝑎 ∈ V ↦ ∅)
8		pred0 6180	. . . . . . 7 ⊢ Pred(𝑅, ∅, 𝑋) = ∅
9		rdgeq12 8051	. . . . . . 7 ⊢ (((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)) = (𝑎 ∈ V ↦ ∅) ∧ Pred(𝑅, ∅, 𝑋) = ∅) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) = rec((𝑎 ∈ V ↦ ∅), ∅))
10	7, 8, 9	mp2an 690	. . . . . 6 ⊢ rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) = rec((𝑎 ∈ V ↦ ∅), ∅)
11	10	reseq1i 5851	. . . . 5 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)
12	11	fveq1i 6673	. . . 4 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖)
13		nn0suc 7608	. . . . 5 ⊢ (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
14		fveq2 6672	. . . . . . 7 ⊢ (𝑖 = ∅ → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘∅))
15		0ex 5213	. . . . . . . 8 ⊢ ∅ ∈ V
16		fr0g 8073	. . . . . . . 8 ⊢ (∅ ∈ V → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘∅) = ∅)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘∅) = ∅
18	14, 17	syl6eq 2874	. . . . . 6 ⊢ (𝑖 = ∅ → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅)
19		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑎∅
20		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑎𝑗
21		eqid 2823	. . . . . . . . . 10 ⊢ (rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω) = (rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)
22		eqidd 2824	. . . . . . . . . 10 ⊢ (𝑎 = ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑗) → ∅ = ∅)
23	19, 20, 19, 21, 22	frsucmpt 8075	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ ∅ ∈ V) → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘suc 𝑗) = ∅)
24	15, 23	mpan2 689	. . . . . . . 8 ⊢ (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘suc 𝑗) = ∅)
25		fveqeq2 6681	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → (((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅ ↔ ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘suc 𝑗) = ∅))
26	24, 25	syl5ibrcom 249	. . . . . . 7 ⊢ (𝑗 ∈ ω → (𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅))
27	26	rexlimiv 3282	. . . . . 6 ⊢ (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅)
28	18, 27	jaoi 853	. . . . 5 ⊢ ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅)
29	13, 28	syl 17	. . . 4 ⊢ (𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅)
30	12, 29	syl5eq 2870	. . 3 ⊢ (𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖) = ∅)
31	30	iuneq2i 4942	. 2 ⊢ ∪ 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖) = ∪ 𝑖 ∈ ω ∅
32		iun0 4987	. 2 ⊢ ∪ 𝑖 ∈ ω ∅ = ∅
33	1, 31, 32	3eqtri 2850	1 ⊢ TrPred(𝑅, ∅, 𝑋) = ∅

Syntax hints:   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496  ∅c0 4293  ∪ ciun 4921   ↦ cmpt 5148   ↾ cres 5559  Predcpred 6149  suc csuc 6195  ‘cfv 6357  ωcom 7582  reccrdg 8047  TrPredctrpred 33058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-trpred 33059

This theorem is referenced by: (None)