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Mirrors > Home > MPE Home > Th. List > rankr1ag | Structured version Visualization version GIF version |
Description: A version of rankr1a 9253 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 9215 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵) | |
2 | r1funlim 9183 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
3 | 2 | simpri 486 | . . . . . . 7 ⊢ Lim dom 𝑅1 |
4 | limord 6243 | . . . . . . 7 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ Ord dom 𝑅1 |
6 | ordelord 6206 | . . . . . 6 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → Ord 𝐵) | |
7 | 5, 6 | mpan 686 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → Ord 𝐵) |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → Ord 𝐵) |
9 | ordsucss 7522 | . . . 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 9217 | . . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
12 | elfvdm 6695 | . . . . 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 9196 | . . . 4 ⊢ ((suc (rank‘𝐴) ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (suc (rank‘𝐴) ⊆ 𝐵 → (𝑅1‘suc (rank‘𝐴)) ⊆ (𝑅1‘𝐵))) | |
15 | 13, 14 | sylan 580 | . . 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 3958 | . . . 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 227 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ∪ cuni 4830 dom cdm 5548 “ cima 5551 Ord word 6183 Oncon0 6184 Lim wlim 6185 suc csuc 6186 Fun wfun 6342 ‘cfv 6348 𝑅1cr1 9179 rankcrnk 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 df-rank 9182 |
This theorem is referenced by: rankr1bg 9220 rankr1clem 9237 rankr1c 9238 rankval3b 9243 onssr1 9248 r1pw 9262 r1pwcl 9264 hsmexlem6 9841 r1limwun 10146 inatsk 10188 grur1 10230 |
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