Theorem tz9.1c 9651

Description: Alternate expression for the existence of transitive closures tz9.1 9650: 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 2736	. . . . 5 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
3		eqid 2736	. . . . 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 9649	. . . 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 1149	. . . 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 9564	. . . . . 6 ⊢ ω ∈ V
7		fvex 6853	. . . . . 6 ⊢ ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
8	6, 7	iunex 7921	. . . . 5 ⊢ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
9		sseq2 3948	. . . . . 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 5199	. . . . . 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 633	. . . . 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 3548	. . . 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 4325	. . 3 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
15	13, 14	mpbir 231	. 2 ⊢ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅
16		intex 5285	. 2 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
17	15, 16	mpbi 230	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714   ≠ wne 2932  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  ∪ cuni 4850  ∩ cint 4889  ∪ ciun 4933   ↦ cmpt 5166  Tr wtr 5192   ↾ cres 5633  ‘cfv 6498  ωcom 7817  reccrdg 8348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349

This theorem is referenced by:  tcvalg  9657