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Theorem rankbnd2 9274
Description: The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.)
Hypothesis
Ref Expression
rankr1b.1 𝐴 ∈ V
Assertion
Ref Expression
rankbnd2 (𝐵 ∈ On → (∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rankbnd2
StepHypRef Expression
1 rankuni 9268 . . . . 5 (rank‘ 𝐴) = (rank‘𝐴)
2 rankr1b.1 . . . . . 6 𝐴 ∈ V
32rankuni2 9260 . . . . 5 (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥)
41, 3eqtr3i 2845 . . . 4 (rank‘𝐴) = 𝑥𝐴 (rank‘𝑥)
54sseq1i 3971 . . 3 ( (rank‘𝐴) ⊆ 𝐵 𝑥𝐴 (rank‘𝑥) ⊆ 𝐵)
6 iunss 4943 . . 3 ( 𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵)
75, 6bitr2i 278 . 2 (∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 (rank‘𝐴) ⊆ 𝐵)
8 rankon 9200 . . . 4 (rank‘𝐴) ∈ On
98onssi 7528 . . 3 (rank‘𝐴) ⊆ On
10 eloni 6175 . . 3 (𝐵 ∈ On → Ord 𝐵)
11 ordunisssuc 6267 . . 3 (((rank‘𝐴) ⊆ On ∧ Ord 𝐵) → ( (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
129, 10, 11sylancr 589 . 2 (𝐵 ∈ On → ( (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
137, 12syl5bb 285 1 (𝐵 ∈ On → (∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wcel 2114  wral 3125  Vcvv 3473  wss 3912   cuni 4812   ciun 4893  Ord word 6164  Oncon0 6165  suc csuc 6167  cfv 6329  rankcrnk 9168
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437  ax-reg 9032  ax-inf2 9080
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-int 4851  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6122  df-ord 6168  df-on 6169  df-lim 6170  df-suc 6171  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-om 7557  df-wrecs 7923  df-recs 7984  df-rdg 8022  df-r1 9169  df-rank 9170
This theorem is referenced by: (None)
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