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Mirrors > Home > MPE Home > Th. List > rankelb | Structured version Visualization version GIF version |
Description: The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankelb | ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1elssi 8706 | . . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ ∪ (𝑅1 “ On)) | |
2 | 1 | sseld 3635 | . . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ (𝑅1 “ On))) |
3 | rankidn 8723 | . . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))) | |
4 | 2, 3 | syl6 35 | . . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))) |
5 | 4 | imp 444 | . . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))) |
6 | rankon 8696 | . . . . 5 ⊢ (rank‘𝐵) ∈ On | |
7 | rankon 8696 | . . . . 5 ⊢ (rank‘𝐴) ∈ On | |
8 | ontri1 5795 | . . . . 5 ⊢ (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))) | |
9 | 6, 7, 8 | mp2an 708 | . . . 4 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)) |
10 | rankdmr1 8702 | . . . . . 6 ⊢ (rank‘𝐵) ∈ dom 𝑅1 | |
11 | rankdmr1 8702 | . . . . . 6 ⊢ (rank‘𝐴) ∈ dom 𝑅1 | |
12 | r1ord3g 8680 | . . . . . 6 ⊢ (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))) | |
13 | 10, 11, 12 | mp2an 708 | . . . . 5 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))) |
14 | r1rankidb 8705 | . . . . . . 7 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵))) | |
15 | 14 | sselda 3636 | . . . . . 6 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵))) |
16 | ssel 3630 | . . . . . 6 ⊢ ((𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ (𝑅1‘(rank‘𝐵)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴)))) | |
17 | 15, 16 | syl5com 31 | . . . . 5 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ((𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴)))) |
18 | 13, 17 | syl5 34 | . . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ((rank‘𝐵) ⊆ (rank‘𝐴) → 𝐴 ∈ (𝑅1‘(rank‘𝐴)))) |
19 | 9, 18 | syl5bir 233 | . . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → (¬ (rank‘𝐴) ∈ (rank‘𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐴)))) |
20 | 5, 19 | mt3d 140 | . 2 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵)) |
21 | 20 | ex 449 | 1 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2030 ⊆ wss 3607 ∪ cuni 4468 dom cdm 5143 “ cima 5146 Oncon0 5761 ‘cfv 5926 𝑅1cr1 8663 rankcrnk 8664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-r1 8665 df-rank 8666 |
This theorem is referenced by: wfelirr 8726 rankval3b 8727 rankel 8740 rankunb 8751 rankuni2b 8754 rankcf 9637 |
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