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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 8546 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
2 | elfvdm 6130 | . . . 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 8512 | . . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpri 477 | . . . 4 ⊢ Lim dom 𝑅1 |
6 | limsuc 6941 | . . . 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 223 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
9 | rankvaln 8545 | . . 3 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅) | |
10 | limomss 6962 | . . . . 5 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
11 | 5, 10 | ax-mp 5 | . . . 4 ⊢ ω ⊆ dom 𝑅1 |
12 | peano1 6977 | . . . 4 ⊢ ∅ ∈ ω | |
13 | 11, 12 | sselii 3565 | . . 3 ⊢ ∅ ∈ dom 𝑅1 |
14 | 9, 13 | syl6eqel 2696 | . 2 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
15 | 8, 14 | pm2.61i 175 | 1 ⊢ (rank‘𝐴) ∈ dom 𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 dom cdm 5038 “ cima 5041 Oncon0 5640 Lim wlim 5641 suc csuc 5642 Fun wfun 5798 ‘cfv 5804 ωcom 6957 𝑅1cr1 8508 rankcrnk 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-rank 8511 |
This theorem is referenced by: r1rankidb 8550 pwwf 8553 unwf 8556 uniwf 8565 rankr1c 8567 rankelb 8570 rankval3b 8572 rankonid 8575 rankssb 8594 rankr1id 8608 |
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