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Mirrors > Home > MPE Home > Th. List > tz9.1c | Structured version Visualization version GIF version |
Description: Alternate expression for the existence of transitive closures tz9.1 9155: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.) |
Ref | Expression |
---|---|
tz9.1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
tz9.1c | ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tz9.1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | eqid 2798 | . . . . 5 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) | |
3 | eqid 2798 | . . . . 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 9154 | . . . 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 1145 | . . . 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 9090 | . . . . . 6 ⊢ ω ∈ V | |
7 | fvex 6658 | . . . . . 6 ⊢ ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V | |
8 | 6, 7 | iunex 7651 | . . . . 5 ⊢ ∪ 𝑤 ∈ ω ((rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)‘𝑤) ∈ V |
9 | sseq2 3941 | . . . . . 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 5142 | . . . . . 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 3555 | . . . 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 4290 | . . 3 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)) | |
15 | 13, 14 | mpbir 234 | . 2 ⊢ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ |
16 | intex 5204 | . 2 ⊢ ({𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ≠ ∅ ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) | |
17 | 15, 16 | mpbi 233 | 1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ≠ wne 2987 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ∩ cint 4838 ∪ ciun 4881 ↦ cmpt 5110 Tr wtr 5136 ↾ cres 5521 ‘cfv 6324 ωcom 7560 reccrdg 8028 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 |
This theorem is referenced by: tcvalg 9164 |
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