Theorem tz9.1c 9687

Description: Alternate expression for the existence of transitive closures tz9.1 9686: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)

Hypothesis

Ref	Expression
tz9.1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tz9.1c	⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

Distinct variable group:   𝑥,𝐴

Proof of Theorem tz9.1c

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

Step	Hyp	Ref	Expression
1		tz9.1.1	. . . . 5 ⊢ 𝐴 ∈ V
2		eqid 2765	. . . . 5 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
3		eqid 2765	. . . . 5 ⊢ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) = ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤)
4	1, 2, 3	trcl 9685	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∧ Tr ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ⊆ 𝑥))
5		3simpa 1164	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∧ Tr ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ⊆ 𝑥)) → (𝐴 ⊆ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∧ Tr ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤)))
6		omex 9600	. . . . . 6 ⊢ ω ∈ V
7		fvex 6884	. . . . . 6 ⊢ ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
8	6, 7	iunex 7953	. . . . 5 ⊢ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
9		sseq2 3965	. . . . . 6 ⊢ (𝑥 = ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤)))
10		treq 5219	. . . . . 6 ⊢ (𝑥 = ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) → (Tr 𝑥 ↔ Tr ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤)))
11	9, 10	anbi12d 643	. . . . 5 ⊢ (𝑥 = ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∧ Tr ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤))))
12	8, 11	spcev 3568	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∧ Tr ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤)) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
13	4, 5, 12	mp2b 10	. . 3 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)
14		abn0 4341	. . 3 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
15	13, 14	mpbir 234	. 2 ⊢ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅
16		intex 5305	. 2 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
17	15, 16	mpbi 233	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743   ≠ wne 2960  Vcvv 3457   ∪ cun 3905   ⊆ wss 3907  ∅c0 4288  ∪ cuni 4868  ∩ cint 4908  ∪ ciun 4952   ↦ cmpt 5186  Tr wtr 5212   ↾ cres 5654  ‘cfv 6525  ωcom 7850  reccrdg 8384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385

This theorem is referenced by:  tcvalg  9693