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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 4844 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | rankr1b.1 | . . . . . . . 8 ⊢ 𝐴 ∈ V | |
3 | 2 | uniex 7507 | . . . . . . 7 ⊢ ∪ 𝐴 ∈ V |
4 | 3 | pwex 5258 | . . . . . 6 ⊢ 𝒫 ∪ 𝐴 ∈ V |
5 | 4 | rankss 9430 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 ∪ 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴)) |
6 | 1, 5 | ax-mp 5 | . . . 4 ⊢ (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴) |
7 | 3 | rankpw 9424 | . . . 4 ⊢ (rank‘𝒫 ∪ 𝐴) = suc (rank‘∪ 𝐴) |
8 | 6, 7 | sseqtri 3923 | . . 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 9420 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)) |
11 | eleq1 2818 | . . . . 5 ⊢ ((rank‘𝑥) = (rank‘∪ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘∪ 𝐴) ∈ (rank‘𝐴))) | |
12 | 10, 11 | syl5ibcom 248 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴))) |
13 | 12 | rexlimiv 3189 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴)) |
14 | rankon 9376 | . . . 4 ⊢ (rank‘∪ 𝐴) ∈ On | |
15 | rankon 9376 | . . . 4 ⊢ (rank‘𝐴) ∈ On | |
16 | 14, 15 | onsucssi 7598 | . . 3 ⊢ ((rank‘∪ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
17 | 13, 16 | sylib 221 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
18 | 9, 17 | eqssd 3904 | 1 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ⊆ wss 3853 𝒫 cpw 4499 ∪ cuni 4805 suc csuc 6193 ‘cfv 6358 rankcrnk 9344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-reg 9186 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-r1 9345 df-rank 9346 |
This theorem is referenced by: (None) |
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