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Mirrors > Home > MPE Home > Th. List > rankr1ag | Structured version Visualization version GIF version |
Description: A version of rankr1a 9258 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 9220 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵) | |
2 | r1funlim 9188 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
3 | 2 | simpri 488 | . . . . . . 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 688 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → Ord 𝐵) |
8 | 7 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → Ord 𝐵) |
9 | ordsucss 7526 | . . . 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 9222 | . . . . 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 9201 | . . . 4 ⊢ ((suc (rank‘𝐴) ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (suc (rank‘𝐴) ⊆ 𝐵 → (𝑅1‘suc (rank‘𝐴)) ⊆ (𝑅1‘𝐵))) | |
15 | 13, 14 | sylan 582 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (suc (rank‘𝐴) ⊆ 𝐵 → (𝑅1‘suc (rank‘𝐴)) ⊆ (𝑅1‘𝐵))) |
16 | 11 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) |
17 | ssel 3954 | . . . 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 228 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ⊆ wss 3929 ∪ cuni 4831 dom cdm 5548 “ cima 5551 Ord word 6183 Oncon0 6184 Lim wlim 6185 suc csuc 6186 Fun wfun 6342 ‘cfv 6348 𝑅1cr1 9184 rankcrnk 9185 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-r1 9186 df-rank 9187 |
This theorem is referenced by: rankr1bg 9225 rankr1clem 9242 rankr1c 9243 rankval3b 9248 onssr1 9253 r1pw 9267 r1pwcl 9269 hsmexlem6 9846 r1limwun 10151 inatsk 10193 grur1 10235 |
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