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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 4825 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | rankr1b.1 | . . . . . . . 8 ⊢ 𝐴 ∈ V | |
3 | 2 | uniex 6851 | . . . . . . 7 ⊢ ∪ 𝐴 ∈ V |
4 | 3 | pwex 4774 | . . . . . 6 ⊢ 𝒫 ∪ 𝐴 ∈ V |
5 | 4 | rankss 8595 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 ∪ 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴)) |
6 | 1, 5 | ax-mp 5 | . . . 4 ⊢ (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴) |
7 | 3 | rankpw 8589 | . . . 4 ⊢ (rank‘𝒫 ∪ 𝐴) = suc (rank‘∪ 𝐴) |
8 | 6, 7 | sseqtri 3600 | . . 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 8585 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)) |
11 | eleq1 2676 | . . . . 5 ⊢ ((rank‘𝑥) = (rank‘∪ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘∪ 𝐴) ∈ (rank‘𝐴))) | |
12 | 10, 11 | syl5ibcom 234 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴))) |
13 | 12 | rexlimiv 3009 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴)) |
14 | rankon 8541 | . . . 4 ⊢ (rank‘∪ 𝐴) ∈ On | |
15 | rankon 8541 | . . . 4 ⊢ (rank‘𝐴) ∈ On | |
16 | 14, 15 | onsucssi 6933 | . . 3 ⊢ ((rank‘∪ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
17 | 13, 16 | sylib 207 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴)) |
18 | 9, 17 | eqssd 3585 | 1 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 suc csuc 5642 ‘cfv 5804 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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
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: (None) |
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