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Mirrors > Home > MPE Home > Th. List > rankdmr1 | Structured version Visualization version GIF version |
Description: A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankdmr1 | ⊢ (rank‘𝐴) ∈ dom 𝑅1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankidb 8838 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
2 | elfvdm 6382 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → suc (rank‘𝐴) ∈ dom 𝑅1) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) ∈ dom 𝑅1) |
4 | r1funlim 8804 | . . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpri 481 | . . . 4 ⊢ Lim dom 𝑅1 |
6 | limsuc 7215 | . . . 4 ⊢ (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1) |
8 | 3, 7 | sylibr 224 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
9 | rankvaln 8837 | . . 3 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅) | |
10 | limomss 7236 | . . . . 5 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
11 | 5, 10 | ax-mp 5 | . . . 4 ⊢ ω ⊆ dom 𝑅1 |
12 | peano1 7251 | . . . 4 ⊢ ∅ ∈ ω | |
13 | 11, 12 | sselii 3741 | . . 3 ⊢ ∅ ∈ dom 𝑅1 |
14 | 9, 13 | syl6eqel 2847 | . 2 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
15 | 8, 14 | pm2.61i 176 | 1 ⊢ (rank‘𝐴) ∈ dom 𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∈ wcel 2139 ⊆ wss 3715 ∅c0 4058 ∪ cuni 4588 dom cdm 5266 “ cima 5269 Oncon0 5884 Lim wlim 5885 suc csuc 5886 Fun wfun 6043 ‘cfv 6049 ωcom 7231 𝑅1cr1 8800 rankcrnk 8801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-r1 8802 df-rank 8803 |
This theorem is referenced by: r1rankidb 8842 pwwf 8845 unwf 8848 uniwf 8857 rankr1c 8859 rankelb 8862 rankval3b 8864 rankonid 8867 rankssb 8886 rankr1id 8900 |
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