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Mirrors > Home > MPE Home > Th. List > ranksuc | Structured version Visualization version GIF version |
Description: The rank of a successor. (Contributed by NM, 18-Sep-2006.) |
Ref | Expression |
---|---|
rankr1b.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ranksuc | ⊢ (rank‘suc 𝐴) = suc (rank‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 6367 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | fveq2i 6891 | . 2 ⊢ (rank‘suc 𝐴) = (rank‘(𝐴 ∪ {𝐴})) |
3 | rankr1b.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | snex 5430 | . . . 4 ⊢ {𝐴} ∈ V | |
5 | 3, 4 | rankun 9847 | . . 3 ⊢ (rank‘(𝐴 ∪ {𝐴})) = ((rank‘𝐴) ∪ (rank‘{𝐴})) |
6 | 3 | ranksn 9845 | . . . . 5 ⊢ (rank‘{𝐴}) = suc (rank‘𝐴) |
7 | 6 | uneq2i 4159 | . . . 4 ⊢ ((rank‘𝐴) ∪ (rank‘{𝐴})) = ((rank‘𝐴) ∪ suc (rank‘𝐴)) |
8 | sssucid 6441 | . . . . 5 ⊢ (rank‘𝐴) ⊆ suc (rank‘𝐴) | |
9 | ssequn1 4179 | . . . . 5 ⊢ ((rank‘𝐴) ⊆ suc (rank‘𝐴) ↔ ((rank‘𝐴) ∪ suc (rank‘𝐴)) = suc (rank‘𝐴)) | |
10 | 8, 9 | mpbi 229 | . . . 4 ⊢ ((rank‘𝐴) ∪ suc (rank‘𝐴)) = suc (rank‘𝐴) |
11 | 7, 10 | eqtri 2761 | . . 3 ⊢ ((rank‘𝐴) ∪ (rank‘{𝐴})) = suc (rank‘𝐴) |
12 | 5, 11 | eqtri 2761 | . 2 ⊢ (rank‘(𝐴 ∪ {𝐴})) = suc (rank‘𝐴) |
13 | 2, 12 | eqtri 2761 | 1 ⊢ (rank‘suc 𝐴) = suc (rank‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3945 ⊆ wss 3947 {csn 4627 suc csuc 6363 ‘cfv 6540 rankcrnk 9754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-reg 9583 ax-inf2 9632 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-r1 9755 df-rank 9756 |
This theorem is referenced by: (None) |
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