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Mirrors > Home > MPE Home > Th. List > r1rankidb | Structured version Visualization version GIF version |
Description: Any set is a subset of the hierarchy of its rank. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
r1rankidb | ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3872 | . 2 ⊢ (rank‘𝐴) ⊆ (rank‘𝐴) | |
2 | rankdmr1 9022 | . . 3 ⊢ (rank‘𝐴) ∈ dom 𝑅1 | |
3 | rankr1bg 9024 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ (rank‘𝐴) ⊆ (rank‘𝐴))) | |
4 | 2, 3 | mpan2 679 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ (rank‘𝐴) ⊆ (rank‘𝐴))) |
5 | 1, 4 | mpbiri 250 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2051 ⊆ wss 3822 ∪ cuni 4708 dom cdm 5403 “ cima 5406 Oncon0 6026 ‘cfv 6185 𝑅1cr1 8983 rankcrnk 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-r1 8985 df-rank 8986 |
This theorem is referenced by: pwwf 9028 unwf 9031 rankpwi 9044 rankelb 9045 rankssb 9069 r1rankid 9080 tcrank 9105 |
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