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Theorem rankc2 9872
Description: A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
Hypothesis
Ref Expression
rankr1b.1 𝐴 ∈ V
Assertion
Ref Expression
rankc2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) = suc (rank‘ 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankc2
StepHypRef Expression
1 pwuni 4949 . . . . 5 𝐴 ⊆ 𝒫 𝐴
2 rankr1b.1 . . . . . . . 8 𝐴 ∈ V
32uniex 7735 . . . . . . 7 𝐴 ∈ V
43pwex 5378 . . . . . 6 𝒫 𝐴 ∈ V
54rankss 9850 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 𝐴))
61, 5ax-mp 5 . . . 4 (rank‘𝐴) ⊆ (rank‘𝒫 𝐴)
73rankpw 9844 . . . 4 (rank‘𝒫 𝐴) = suc (rank‘ 𝐴)
86, 7sseqtri 4018 . . 3 (rank‘𝐴) ⊆ suc (rank‘ 𝐴)
98a1i 11 . 2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) ⊆ suc (rank‘ 𝐴))
102rankel 9840 . . . . 5 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
11 eleq1 2820 . . . . 5 ((rank‘𝑥) = (rank‘ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘ 𝐴) ∈ (rank‘𝐴)))
1210, 11syl5ibcom 244 . . . 4 (𝑥𝐴 → ((rank‘𝑥) = (rank‘ 𝐴) → (rank‘ 𝐴) ∈ (rank‘𝐴)))
1312rexlimiv 3147 . . 3 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘ 𝐴) ∈ (rank‘𝐴))
14 rankon 9796 . . . 4 (rank‘ 𝐴) ∈ On
15 rankon 9796 . . . 4 (rank‘𝐴) ∈ On
1614, 15onsucssi 7834 . . 3 ((rank‘ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘ 𝐴) ⊆ (rank‘𝐴))
1713, 16sylib 217 . 2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → suc (rank‘ 𝐴) ⊆ (rank‘𝐴))
189, 17eqssd 3999 1 (∃𝑥𝐴 (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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