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Mirrors > Home > MPE Home > Th. List > rankc2 | Structured version Visualization version GIF version |
Description: A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
Ref | Expression |
---|---|
rankr1b.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rankc2 | ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4787 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | rankr1b.1 | . . . . . . . 8 ⊢ 𝐴 ∈ V | |
3 | 2 | uniex 7330 | . . . . . . 7 ⊢ ∪ 𝐴 ∈ V |
4 | 3 | pwex 5179 | . . . . . 6 ⊢ 𝒫 ∪ 𝐴 ∈ V |
5 | 4 | rankss 9131 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 ∪ 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴)) |
6 | 1, 5 | ax-mp 5 | . . . 4 ⊢ (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴) |
7 | 3 | rankpw 9125 | . . . 4 ⊢ (rank‘𝒫 ∪ 𝐴) = suc (rank‘∪ 𝐴) |
8 | 6, 7 | sseqtri 3930 | . . 3 ⊢ (rank‘𝐴) ⊆ suc (rank‘∪ 𝐴) |
9 | 8 | a1i 11 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) ⊆ suc (rank‘∪ 𝐴)) |
10 | 2 | rankel 9121 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)) |
11 | eleq1 2872 | . . . . 5 ⊢ ((rank‘𝑥) = (rank‘∪ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘∪ 𝐴) ∈ (rank‘𝐴))) | |
12 | 10, 11 | syl5ibcom 246 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴))) |
13 | 12 | rexlimiv 3245 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴)) |
14 | rankon 9077 | . . . 4 ⊢ (rank‘∪ 𝐴) ∈ On | |
15 | rankon 9077 | . . . 4 ⊢ (rank‘𝐴) ∈ On | |
16 | 14, 15 | onsucssi 7419 | . . 3 ⊢ ((rank‘∪ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
17 | 13, 16 | sylib 219 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
18 | 9, 17 | eqssd 3912 | 1 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ∃wrex 3108 Vcvv 3440 ⊆ wss 3865 𝒫 cpw 4459 ∪ cuni 4751 suc csuc 6075 ‘cfv 6232 rankcrnk 9045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-reg 8909 ax-inf2 8957 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-r1 9046 df-rank 9047 |
This theorem is referenced by: (None) |
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