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Mirrors > Home > MPE Home > Th. List > rankval3 | Structured version Visualization version GIF version |
Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankval3.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rankval3 | ⊢ (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankval3.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | unir1 9091 | . . 3 ⊢ ∪ (𝑅1 “ On) = V | |
3 | 1, 2 | eleqtrri 2881 | . 2 ⊢ 𝐴 ∈ ∪ (𝑅1 “ On) |
4 | rankval3b 9104 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥}) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2080 ∀wral 3104 {crab 3108 Vcvv 3436 ∪ cuni 4747 ∩ cint 4784 “ cima 5449 Oncon0 6069 ‘cfv 6228 𝑅1cr1 9040 rankcrnk 9041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-reg 8905 ax-inf2 8953 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-r1 9042 df-rank 9043 |
This theorem is referenced by: (None) |
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