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Mirrors > Home > MPE Home > Th. List > rankelop | Structured version Visualization version GIF version |
Description: Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.) |
Ref | Expression |
---|---|
rankelun.1 | ⊢ 𝐴 ∈ V |
rankelun.2 | ⊢ 𝐵 ∈ V |
rankelun.3 | ⊢ 𝐶 ∈ V |
rankelun.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
rankelop | ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘〈𝐴, 𝐵〉) ∈ (rank‘〈𝐶, 𝐷〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankelun.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | rankelun.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | rankelun.3 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | rankelun.4 | . . . 4 ⊢ 𝐷 ∈ V | |
5 | 1, 2, 3, 4 | rankelpr 9805 | . . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷})) |
6 | rankon 9727 | . . . . 5 ⊢ (rank‘{𝐶, 𝐷}) ∈ On | |
7 | 6 | onordi 6425 | . . . 4 ⊢ Ord (rank‘{𝐶, 𝐷}) |
8 | ordsucelsuc 7753 | . . . 4 ⊢ (Ord (rank‘{𝐶, 𝐷}) → ((rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}) ↔ suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷}))) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ ((rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}) ↔ suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷})) |
10 | 5, 9 | sylib 217 | . 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷})) |
11 | 1, 2 | rankop 9790 | . . 3 ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
12 | 1, 2 | rankpr 9789 | . . . 4 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
13 | suceq 6381 | . . . 4 ⊢ ((rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) → suc (rank‘{𝐴, 𝐵}) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ suc (rank‘{𝐴, 𝐵}) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
15 | 11, 14 | eqtr4i 2767 | . 2 ⊢ (rank‘〈𝐴, 𝐵〉) = suc (rank‘{𝐴, 𝐵}) |
16 | 3, 4 | rankop 9790 | . . 3 ⊢ (rank‘〈𝐶, 𝐷〉) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷)) |
17 | 3, 4 | rankpr 9789 | . . . 4 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷)) |
18 | suceq 6381 | . . . 4 ⊢ ((rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷)) → suc (rank‘{𝐶, 𝐷}) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷))) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ suc (rank‘{𝐶, 𝐷}) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷)) |
20 | 16, 19 | eqtr4i 2767 | . 2 ⊢ (rank‘〈𝐶, 𝐷〉) = suc (rank‘{𝐶, 𝐷}) |
21 | 10, 15, 20 | 3eltr4g 2855 | 1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘〈𝐴, 𝐵〉) ∈ (rank‘〈𝐶, 𝐷〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∪ cun 3906 {cpr 4586 〈cop 4590 Ord word 6314 suc csuc 6317 ‘cfv 6493 rankcrnk 9695 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-reg 9524 ax-inf2 9573 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-r1 9696 df-rank 9697 |
This theorem is referenced by: (None) |
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