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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 4949 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | rankr1b.1 | . . . . . . . 8 ⊢ 𝐴 ∈ V | |
3 | 2 | uniex 7735 | . . . . . . 7 ⊢ ∪ 𝐴 ∈ V |
4 | 3 | pwex 5378 | . . . . . 6 ⊢ 𝒫 ∪ 𝐴 ∈ V |
5 | 4 | rankss 9850 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 ∪ 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴)) |
6 | 1, 5 | ax-mp 5 | . . . 4 ⊢ (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴) |
7 | 3 | rankpw 9844 | . . . 4 ⊢ (rank‘𝒫 ∪ 𝐴) = suc (rank‘∪ 𝐴) |
8 | 6, 7 | sseqtri 4018 | . . 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 9840 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)) |
11 | eleq1 2820 | . . . . 5 ⊢ ((rank‘𝑥) = (rank‘∪ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘∪ 𝐴) ∈ (rank‘𝐴))) | |
12 | 10, 11 | syl5ibcom 244 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴))) |
13 | 12 | rexlimiv 3147 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴)) |
14 | rankon 9796 | . . . 4 ⊢ (rank‘∪ 𝐴) ∈ On | |
15 | rankon 9796 | . . . 4 ⊢ (rank‘𝐴) ∈ On | |
16 | 14, 15 | onsucssi 7834 | . . 3 ⊢ ((rank‘∪ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
17 | 13, 16 | sylib 217 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
18 | 9, 17 | eqssd 3999 | 1 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 Vcvv 3473 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 suc csuc 6366 ‘cfv 6543 rankcrnk 9764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-reg 9593 ax-inf2 9642 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-r1 9765 df-rank 9766 |
This theorem is referenced by: (None) |
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