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| 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 9913 | . . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷})) | 
| 6 | rankon 9835 | . . . . 5 ⊢ (rank‘{𝐶, 𝐷}) ∈ On | |
| 7 | 6 | onordi 6495 | . . . 4 ⊢ Ord (rank‘{𝐶, 𝐷}) | 
| 8 | ordsucelsuc 7842 | . . . 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 218 | . 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷})) | 
| 11 | 1, 2 | rankop 9898 | . . 3 ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) | 
| 12 | 1, 2 | rankpr 9897 | . . . 4 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) | 
| 13 | suceq 6450 | . . . 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 2768 | . 2 ⊢ (rank‘〈𝐴, 𝐵〉) = suc (rank‘{𝐴, 𝐵}) | 
| 16 | 3, 4 | rankop 9898 | . . 3 ⊢ (rank‘〈𝐶, 𝐷〉) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷)) | 
| 17 | 3, 4 | rankpr 9897 | . . . 4 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷)) | 
| 18 | suceq 6450 | . . . 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 2768 | . 2 ⊢ (rank‘〈𝐶, 𝐷〉) = suc (rank‘{𝐶, 𝐷}) | 
| 21 | 10, 15, 20 | 3eltr4g 2858 | 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {cpr 4628 〈cop 4632 Ord word 6383 suc csuc 6386 ‘cfv 6561 rankcrnk 9803 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 df-rank 9805 | 
| This theorem is referenced by: (None) | 
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