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Mirrors > Home > MPE Home > Th. List > rankop | Structured version Visualization version GIF version |
Description: The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
ranksn.1 | ⊢ 𝐴 ∈ V |
rankun.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rankop | ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ranksn.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | unir1 9804 | . . 3 ⊢ ∪ (𝑅1 “ On) = V | |
3 | 1, 2 | eleqtrri 2832 | . 2 ⊢ 𝐴 ∈ ∪ (𝑅1 “ On) |
4 | rankun.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 4, 2 | eleqtrri 2832 | . 2 ⊢ 𝐵 ∈ ∪ (𝑅1 “ On) |
6 | rankopb 9843 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))) | |
7 | 3, 5, 6 | mp2an 690 | 1 ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cun 3945 〈cop 4633 ∪ cuni 4907 “ cima 5678 Oncon0 6361 suc csuc 6363 ‘cfv 6540 𝑅1cr1 9753 rankcrnk 9754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-reg 9583 ax-inf2 9632 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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 7408 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-r1 9755 df-rank 9756 |
This theorem is referenced by: rankelop 9865 |
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