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Mirrors > Home > MPE Home > Th. List > rankr1clem | Structured version Visualization version GIF version |
Description: Lemma for rankr1c 8983. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankr1clem | ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankr1ag 8964 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) | |
2 | 1 | notbid 310 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) |
3 | r1funlim 8928 | . . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
4 | 3 | simpri 481 | . . . . . 6 ⊢ Lim dom 𝑅1 |
5 | limord 6037 | . . . . . 6 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ Ord dom 𝑅1 |
7 | ordelon 6002 | . . . . 5 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) | |
8 | 6, 7 | mpan 680 | . . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On) |
9 | 8 | adantl 475 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) |
10 | rankon 8957 | . . 3 ⊢ (rank‘𝐴) ∈ On | |
11 | ontri1 6012 | . . 3 ⊢ ((𝐵 ∈ On ∧ (rank‘𝐴) ∈ On) → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) | |
12 | 9, 10, 11 | sylancl 580 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) |
13 | 2, 12 | bitr4d 274 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2107 ⊆ wss 3792 ∪ cuni 4673 dom cdm 5357 “ cima 5360 Ord word 5977 Oncon0 5978 Lim wlim 5979 Fun wfun 6131 ‘cfv 6137 𝑅1cr1 8924 rankcrnk 8925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-r1 8926 df-rank 8927 |
This theorem is referenced by: rankr1c 8983 ssrankr1 8997 |
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