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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 9304 | . . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷})) |
6 | rankon 9226 | . . . . 5 ⊢ (rank‘{𝐶, 𝐷}) ∈ On | |
7 | 6 | onordi 6297 | . . . 4 ⊢ Ord (rank‘{𝐶, 𝐷}) |
8 | ordsucelsuc 7539 | . . . 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 220 | . 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷})) |
11 | 1, 2 | rankop 9289 | . . 3 ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
12 | 1, 2 | rankpr 9288 | . . . 4 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
13 | suceq 6258 | . . . 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 2849 | . 2 ⊢ (rank‘〈𝐴, 𝐵〉) = suc (rank‘{𝐴, 𝐵}) |
16 | 3, 4 | rankop 9289 | . . 3 ⊢ (rank‘〈𝐶, 𝐷〉) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷)) |
17 | 3, 4 | rankpr 9288 | . . . 4 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷)) |
18 | suceq 6258 | . . . 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 2849 | . 2 ⊢ (rank‘〈𝐶, 𝐷〉) = suc (rank‘{𝐶, 𝐷}) |
21 | 10, 15, 20 | 3eltr4g 2932 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 {cpr 4571 〈cop 4575 Ord word 6192 suc csuc 6195 ‘cfv 6357 rankcrnk 9194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-r1 9195 df-rank 9196 |
This theorem is referenced by: (None) |
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