Theorem tz9.1c 9682

Description: Alternate expression for the existence of transitive closures tz9.1 9681: 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 2761	. . . . 5 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)
3		eqid 2761	. . . . 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 9680	. . . 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 1160	. . . 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 9595	. . . . . 6 ⊢ ω ∈ V
7		fvex 6876	. . . . . 6 ⊢ ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
8	6, 7	iunex 7945	. . . . 5 ⊢ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V
9		sseq2 3962	. . . . . 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 5213	. . . . . 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 641	. . . . 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 3565	. . . 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 4337	. . 3 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
15	13, 14	mpbir 233	. 2 ⊢ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅
16		intex 5299	. 2 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
17	15, 16	mpbi 232	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ≠ wne 2956  Vcvv 3453   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4864  ∩ cint 4904  ∪ ciun 4948   ↦ cmpt 5180  Tr wtr 5206   ↾ cres 5647  ‘cfv 6517  ωcom 7842  reccrdg 8375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714  ax-inf2 9593

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376

This theorem is referenced by:  tcvalg  9688