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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 9454 | . . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷})) |
6 | rankon 9376 | . . . . 5 ⊢ (rank‘{𝐶, 𝐷}) ∈ On | |
7 | 6 | onordi 6296 | . . . 4 ⊢ Ord (rank‘{𝐶, 𝐷}) |
8 | ordsucelsuc 7579 | . . . 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 221 | . 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷})) |
11 | 1, 2 | rankop 9439 | . . 3 ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
12 | 1, 2 | rankpr 9438 | . . . 4 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) |
13 | suceq 6256 | . . . 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 2762 | . 2 ⊢ (rank‘〈𝐴, 𝐵〉) = suc (rank‘{𝐴, 𝐵}) |
16 | 3, 4 | rankop 9439 | . . 3 ⊢ (rank‘〈𝐶, 𝐷〉) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷)) |
17 | 3, 4 | rankpr 9438 | . . . 4 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷)) |
18 | suceq 6256 | . . . 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 2762 | . 2 ⊢ (rank‘〈𝐶, 𝐷〉) = suc (rank‘{𝐶, 𝐷}) |
21 | 10, 15, 20 | 3eltr4g 2848 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∪ cun 3851 {cpr 4529 〈cop 4533 Ord word 6190 suc csuc 6193 ‘cfv 6358 rankcrnk 9344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-reg 9186 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-r1 9345 df-rank 9346 |
This theorem is referenced by: (None) |
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