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Mirrors > Home > MPE Home > Th. List > rankr1ag | Structured version Visualization version GIF version |
Description: A version of rankr1a 8699 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankr1ag | ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankr1ai 8661 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵) | |
2 | r1funlim 8629 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
3 | 2 | simpri 478 | . . . . . . 7 ⊢ Lim dom 𝑅1 |
4 | limord 5784 | . . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ Ord dom 𝑅1 |
6 | ordelord 5745 | . . . . . 6 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → Ord 𝐵) | |
7 | 5, 6 | mpan 706 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → Ord 𝐵) |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → Ord 𝐵) |
9 | ordsucss 7018 | . . . 4 ⊢ (Ord 𝐵 → ((rank‘𝐴) ∈ 𝐵 → suc (rank‘𝐴) ⊆ 𝐵)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((rank‘𝐴) ∈ 𝐵 → suc (rank‘𝐴) ⊆ 𝐵)) |
11 | rankidb 8663 | . . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
12 | elfvdm 6220 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → suc (rank‘𝐴) ∈ dom 𝑅1) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) ∈ dom 𝑅1) |
14 | r1ord3g 8642 | . . . 4 ⊢ ((suc (rank‘𝐴) ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (suc (rank‘𝐴) ⊆ 𝐵 → (𝑅1‘suc (rank‘𝐴)) ⊆ (𝑅1‘𝐵))) | |
15 | 13, 14 | sylan 488 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (suc (rank‘𝐴) ⊆ 𝐵 → (𝑅1‘suc (rank‘𝐴)) ⊆ (𝑅1‘𝐵))) |
16 | 11 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) |
17 | ssel 3597 | . . . 4 ⊢ ((𝑅1‘suc (rank‘𝐴)) ⊆ (𝑅1‘𝐵) → (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ∈ (𝑅1‘𝐵))) | |
18 | 16, 17 | syl5com 31 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((𝑅1‘suc (rank‘𝐴)) ⊆ (𝑅1‘𝐵) → 𝐴 ∈ (𝑅1‘𝐵))) |
19 | 10, 15, 18 | 3syld 60 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((rank‘𝐴) ∈ 𝐵 → 𝐴 ∈ (𝑅1‘𝐵))) |
20 | 1, 19 | impbid2 216 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ⊆ wss 3574 ∪ cuni 4436 dom cdm 5114 “ cima 5117 Ord word 5722 Oncon0 5723 Lim wlim 5724 suc csuc 5725 Fun wfun 5882 ‘cfv 5888 𝑅1cr1 8625 rankcrnk 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628 |
This theorem is referenced by: rankr1bg 8666 rankr1clem 8683 rankr1c 8684 rankval3b 8689 onssr1 8694 r1pw 8708 r1pwcl 8710 hsmexlem6 9253 r1limwun 9558 inatsk 9600 grur1 9642 |
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