Theorem trpred0 32678

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 32661	. 2 ⊢ TrPred(𝑅, ∅, 𝑋) = ∪ 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖)
2		pred0 6056	. . . . . . . . . . 11 ⊢ Pred(𝑅, ∅, 𝑦) = ∅
3	2	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑎 → Pred(𝑅, ∅, 𝑦) = ∅)
4	3	iuneq2i 4847	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦) = ∪ 𝑦 ∈ 𝑎 ∅
5		iun0 4886	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ 𝑎 ∅ = ∅
6	4, 5	eqtri 2818	. . . . . . . 8 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦) = ∅
7	6	mpteq2i 5055	. . . . . . 7 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)) = (𝑎 ∈ V ↦ ∅)
8		pred0 6056	. . . . . . 7 ⊢ Pred(𝑅, ∅, 𝑋) = ∅
9		rdgeq12 7904	. . . . . . 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 688	. . . . . 6 ⊢ rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) = rec((𝑎 ∈ V ↦ ∅), ∅)
11	10	reseq1i 5733	. . . . 5 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)
12	11	fveq1i 6542	. . . 4 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖)
13		nn0suc 7465	. . . . 5 ⊢ (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
14		fveq2 6541	. . . . . . 7 ⊢ (𝑖 = ∅ → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘∅))
15		0ex 5105	. . . . . . . 8 ⊢ ∅ ∈ V
16		fr0g 7926	. . . . . . . 8 ⊢ (∅ ∈ V → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘∅) = ∅)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘∅) = ∅
18	14, 17	syl6eq 2846	. . . . . 6 ⊢ (𝑖 = ∅ → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅)
19		nfcv 2948	. . . . . . . . . 10 ⊢ Ⅎ𝑎∅
20		nfcv 2948	. . . . . . . . . 10 ⊢ Ⅎ𝑎𝑗
21		eqid 2794	. . . . . . . . . 10 ⊢ (rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω) = (rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)
22		eqidd 2795	. . . . . . . . . 10 ⊢ (𝑎 = ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑗) → ∅ = ∅)
23	19, 20, 19, 21, 22	frsucmpt 7928	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ ∅ ∈ V) → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘suc 𝑗) = ∅)
24	15, 23	mpan2 687	. . . . . . . 8 ⊢ (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘suc 𝑗) = ∅)
25		fveqeq2 6550	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → (((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅ ↔ ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘suc 𝑗) = ∅))
26	24, 25	syl5ibrcom 248	. . . . . . 7 ⊢ (𝑗 ∈ ω → (𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅))
27	26	rexlimiv 3242	. . . . . 6 ⊢ (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ ∅), ∅) ↾ ω)‘𝑖) = ∅)
28	18, 27	jaoi 852	. . . . 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 2842	. . 3 ⊢ (𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖) = ∅)
31	30	iuneq2i 4847	. 2 ⊢ ∪ 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, ∅, 𝑦)), Pred(𝑅, ∅, 𝑋)) ↾ ω)‘𝑖) = ∪ 𝑖 ∈ ω ∅
32		iun0 4886	. 2 ⊢ ∪ 𝑖 ∈ ω ∅ = ∅
33	1, 31, 32	3eqtri 2822	1 ⊢ TrPred(𝑅, ∅, 𝑋) = ∅

Syntax hints:   ∨ wo 842   = wceq 1522   ∈ wcel 2080  ∃wrex 3105  Vcvv 3436  ∅c0 4213  ∪ ciun 4827   ↦ cmpt 5043   ↾ cres 5448  Predcpred 6025  suc csuc 6071  ‘cfv 6228  ωcom 7439  reccrdg 7900  TrPredctrpred 32659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-om 7440  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-trpred 32660

This theorem is referenced by: (None)