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Mirrors > Home > MPE Home > Th. List > r1sssuc | Structured version Visualization version GIF version |
Description: The value of the cumulative hierarchy of sets function is a subset of its value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL, 20-Apr-2011.) |
Ref | Expression |
---|---|
r1sssuc | ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1tr 9416 | . . 3 ⊢ Tr (𝑅1‘𝐴) | |
2 | dftr4 5180 | . . 3 ⊢ (Tr (𝑅1‘𝐴) ↔ (𝑅1‘𝐴) ⊆ 𝒫 (𝑅1‘𝐴)) | |
3 | 1, 2 | mpbi 233 | . 2 ⊢ (𝑅1‘𝐴) ⊆ 𝒫 (𝑅1‘𝐴) |
4 | r1suc 9410 | . 2 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
5 | 3, 4 | sseqtrrid 3968 | 1 ⊢ (𝐴 ∈ On → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3880 𝒫 cpw 4527 Tr wtr 5175 Oncon0 6230 suc csuc 6232 ‘cfv 6397 𝑅1cr1 9402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-om 7663 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-r1 9404 |
This theorem is referenced by: ackbij2lem3 9879 |
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