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Mirrors > Home > MPE Home > Th. List > r10 | Structured version Visualization version GIF version |
Description: Value of the cumulative hierarchy of sets function at ∅. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
Ref | Expression |
---|---|
r10 | ⊢ (𝑅1‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r1 9193 | . . 3 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
2 | 1 | fveq1i 6671 | . 2 ⊢ (𝑅1‘∅) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘∅) |
3 | 0ex 5211 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | rdg0 8057 | . 2 ⊢ (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘∅) = ∅ |
5 | 2, 4 | eqtri 2844 | 1 ⊢ (𝑅1‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3494 ∅c0 4291 𝒫 cpw 4539 ↦ cmpt 5146 ‘cfv 6355 reccrdg 8045 𝑅1cr1 9191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-r1 9193 |
This theorem is referenced by: r1fin 9202 r1tr 9205 r1pwss 9213 r1val1 9215 rankeq0b 9289 ackbij2lem2 9662 ackbij2lem3 9663 wunr1om 10141 r1wunlim 10159 tskr1om 10189 inar1 10197 r1tskina 10204 grur1a 10241 grothomex 10251 rankeq1o 33632 grur1cld 40588 |
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