Theorem tz9.1c 9734

Description: Alternate expression for the existence of transitive closures tz9.1 9733: 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 2760	. . . . 5 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
3		eqid 2760	. . . . 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 9732	. . . 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 1166	. . . 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 9647	. . . . . 6 ⊢ ω ∈ V
7		fvex 6898	. . . . . 6 ⊢ ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
8	6, 7	iunex 7980	. . . . 5 ⊢ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
9		sseq2 3934	. . . . . 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 5218	. . . . . 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 644	. . . . 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 3560	. . . 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 4333	. . 3 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
15	13, 14	mpbir 234	. 2 ⊢ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅
16		intex 5304	. 2 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
17	15, 16	mpbi 233	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  ∪ cuni 4866  ∩ cint 4906  ∪ ciun 4950   ↦ cmpt 5185  Tr wtr 5211   ↾ cres 5652  ‘cfv 6538  ωcom 7877  reccrdg 8420

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-pr 5390  ax-un 7751  ax-inf2 9645

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 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  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 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421

This theorem is used by:  tcvalg  9740