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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 9521 | . . 3 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
2 | 1 | fveq1i 6770 | . 2 ⊢ (𝑅1‘∅) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘∅) |
3 | 0ex 5235 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | rdg0 8241 | . 2 ⊢ (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘∅) = ∅ |
5 | 2, 4 | eqtri 2768 | 1 ⊢ (𝑅1‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3431 ∅c0 4262 𝒫 cpw 4539 ↦ cmpt 5162 ‘cfv 6431 reccrdg 8229 𝑅1cr1 9519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-ov 7272 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-r1 9521 |
This theorem is referenced by: r1fin 9530 r1tr 9533 r1pwss 9541 r1val1 9543 rankeq0b 9617 ackbij2lem2 9995 ackbij2lem3 9996 wunr1om 10474 r1wunlim 10492 tskr1om 10522 inar1 10530 r1tskina 10537 grur1a 10574 grothomex 10584 rankeq1o 34467 grur1cld 41818 |
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