Theorem tfinsuc 4499

Description: The finite T operator over a successor. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
tfinsuc	⊢ ((A ∈ Nn ∧ (A +c 1c) ≠ ∅) → Tfin (A +c 1c) = ( Tfin A +c 1c))

Proof of Theorem tfinsuc

Dummy variables a b x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3560	. . 3 ⊢ ((A +c 1c) ≠ ∅ ↔ ∃a a ∈ (A +c 1c))
2		peano2 4404	. . . . . . . 8 ⊢ (A ∈ Nn → (A +c 1c) ∈ Nn )
3		tfincl 4493	. . . . . . . 8 ⊢ ((A +c 1c) ∈ Nn → Tfin (A +c 1c) ∈ Nn )
4	2, 3	syl 15	. . . . . . 7 ⊢ (A ∈ Nn → Tfin (A +c 1c) ∈ Nn )
5	4	adantr 451	. . . . . 6 ⊢ ((A ∈ Nn ∧ a ∈ (A +c 1c)) → Tfin (A +c 1c) ∈ Nn )
6		tfincl 4493	. . . . . . . 8 ⊢ (A ∈ Nn → Tfin A ∈ Nn )
7		peano2 4404	. . . . . . . 8 ⊢ ( Tfin A ∈ Nn → ( Tfin A +c 1c) ∈ Nn )
8	6, 7	syl 15	. . . . . . 7 ⊢ (A ∈ Nn → ( Tfin A +c 1c) ∈ Nn )
9	8	adantr 451	. . . . . 6 ⊢ ((A ∈ Nn ∧ a ∈ (A +c 1c)) → ( Tfin A +c 1c) ∈ Nn )
10		tfinpw1 4495	. . . . . . 7 ⊢ (((A +c 1c) ∈ Nn ∧ a ∈ (A +c 1c)) → 𝒫1a ∈ Tfin (A +c 1c))
11	2, 10	sylan 457	. . . . . 6 ⊢ ((A ∈ Nn ∧ a ∈ (A +c 1c)) → 𝒫1a ∈ Tfin (A +c 1c))
12		elsuc 4414	. . . . . . . 8 ⊢ (a ∈ (A +c 1c) ↔ ∃b ∈ A ∃x ∈ ∼ ba = (b ∪ {x}))
13		tfinpw1 4495	. . . . . . . . . . . 12 ⊢ ((A ∈ Nn ∧ b ∈ A) → 𝒫1b ∈ Tfin A)
14	13	adantrr 697	. . . . . . . . . . 11 ⊢ ((A ∈ Nn ∧ (b ∈ A ∧ x ∈ ∼ b)) → 𝒫1b ∈ Tfin A)
15		vex 2863	. . . . . . . . . . . . . . 15 ⊢ x ∈ V
16	15	elcompl 3226	. . . . . . . . . . . . . 14 ⊢ (x ∈ ∼ b ↔ ¬ x ∈ b)
17		snelpw1 4147	. . . . . . . . . . . . . 14 ⊢ ({x} ∈ 𝒫1b ↔ x ∈ b)
18	16, 17	xchbinxr 302	. . . . . . . . . . . . 13 ⊢ (x ∈ ∼ b ↔ ¬ {x} ∈ 𝒫1b)
19	18	biimpi 186	. . . . . . . . . . . 12 ⊢ (x ∈ ∼ b → ¬ {x} ∈ 𝒫1b)
20	19	ad2antll 709	. . . . . . . . . . 11 ⊢ ((A ∈ Nn ∧ (b ∈ A ∧ x ∈ ∼ b)) → ¬ {x} ∈ 𝒫1b)
21		snex 4112	. . . . . . . . . . . 12 ⊢ {x} ∈ V
22	21	elsuci 4415	. . . . . . . . . . 11 ⊢ ((𝒫1b ∈ Tfin A ∧ ¬ {x} ∈ 𝒫1b) → (𝒫1b ∪ {{x}}) ∈ ( Tfin A +c 1c))
23	14, 20, 22	syl2anc 642	. . . . . . . . . 10 ⊢ ((A ∈ Nn ∧ (b ∈ A ∧ x ∈ ∼ b)) → (𝒫1b ∪ {{x}}) ∈ ( Tfin A +c 1c))
24		pw1eq 4144	. . . . . . . . . . . 12 ⊢ (a = (b ∪ {x}) → 𝒫1a = 𝒫1(b ∪ {x}))
25		pw1un 4164	. . . . . . . . . . . . 13 ⊢ 𝒫1(b ∪ {x}) = (𝒫1b ∪ 𝒫1{x})
26	15	pw1sn 4166	. . . . . . . . . . . . . 14 ⊢ 𝒫1{x} = {{x}}
27	26	uneq2i 3416	. . . . . . . . . . . . 13 ⊢ (𝒫1b ∪ 𝒫1{x}) = (𝒫1b ∪ {{x}})
28	25, 27	eqtri 2373	. . . . . . . . . . . 12 ⊢ 𝒫1(b ∪ {x}) = (𝒫1b ∪ {{x}})
29	24, 28	syl6eq 2401	. . . . . . . . . . 11 ⊢ (a = (b ∪ {x}) → 𝒫1a = (𝒫1b ∪ {{x}}))
30	29	eleq1d 2419	. . . . . . . . . 10 ⊢ (a = (b ∪ {x}) → (𝒫1a ∈ ( Tfin A +c 1c) ↔ (𝒫1b ∪ {{x}}) ∈ ( Tfin A +c 1c)))
31	23, 30	syl5ibrcom 213	. . . . . . . . 9 ⊢ ((A ∈ Nn ∧ (b ∈ A ∧ x ∈ ∼ b)) → (a = (b ∪ {x}) → 𝒫1a ∈ ( Tfin A +c 1c)))
32	31	rexlimdvva 2746	. . . . . . . 8 ⊢ (A ∈ Nn → (∃b ∈ A ∃x ∈ ∼ ba = (b ∪ {x}) → 𝒫1a ∈ ( Tfin A +c 1c)))
33	12, 32	syl5bi 208	. . . . . . 7 ⊢ (A ∈ Nn → (a ∈ (A +c 1c) → 𝒫1a ∈ ( Tfin A +c 1c)))
34	33	imp 418	. . . . . 6 ⊢ ((A ∈ Nn ∧ a ∈ (A +c 1c)) → 𝒫1a ∈ ( Tfin A +c 1c))
35		nnceleq 4431	. . . . . 6 ⊢ ((( Tfin (A +c 1c) ∈ Nn ∧ ( Tfin A +c 1c) ∈ Nn ) ∧ (𝒫1a ∈ Tfin (A +c 1c) ∧ 𝒫1a ∈ ( Tfin A +c 1c))) → Tfin (A +c 1c) = ( Tfin A +c 1c))
36	5, 9, 11, 34, 35	syl22anc 1183	. . . . 5 ⊢ ((A ∈ Nn ∧ a ∈ (A +c 1c)) → Tfin (A +c 1c) = ( Tfin A +c 1c))
37	36	ex 423	. . . 4 ⊢ (A ∈ Nn → (a ∈ (A +c 1c) → Tfin (A +c 1c) = ( Tfin A +c 1c)))
38	37	exlimdv 1636	. . 3 ⊢ (A ∈ Nn → (∃a a ∈ (A +c 1c) → Tfin (A +c 1c) = ( Tfin A +c 1c)))
39	1, 38	syl5bi 208	. 2 ⊢ (A ∈ Nn → ((A +c 1c) ≠ ∅ → Tfin (A +c 1c) = ( Tfin A +c 1c)))
40	39	imp 418	1 ⊢ ((A ∈ Nn ∧ (A +c 1c) ≠ ∅) → Tfin (A +c 1c) = ( Tfin A +c 1c))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208  ∅c0 3551  {csn 3738  1_cc1c 4135  𝒫₁cpw1 4136   Nn cnnc 4374   +_c cplc 4376   T_fin ctfin 4436

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-tfin 4444

This theorem is used by:  tfin1c  4500  tfinltfinlem1  4501  sfintfin  4533  tfinnn  4535