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Theorem rankbnd2 8615
 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 8609 . . . . 5 (rank‘ 𝐴) = (rank‘𝐴)
2 rankr1b.1 . . . . . 6 𝐴 ∈ V
32rankuni2 8601 . . . . 5 (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥)
41, 3eqtr3i 2634 . . . 4 (rank‘𝐴) = 𝑥𝐴 (rank‘𝑥)
54sseq1i 3592 . . 3 ( (rank‘𝐴) ⊆ 𝐵 𝑥𝐴 (rank‘𝑥) ⊆ 𝐵)
6 iunss 4497 . . 3 ( 𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵)
75, 6bitr2i 264 . 2 (∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 (rank‘𝐴) ⊆ 𝐵)
8 rankon 8541 . . . 4 (rank‘𝐴) ∈ On
98onssi 6929 . . 3 (rank‘𝐴) ⊆ On
10 eloni 5650 . . 3 (𝐵 ∈ On → Ord 𝐵)
11 ordunisssuc 5747 . . 3 (((rank‘𝐴) ⊆ On ∧ Ord 𝐵) → ( (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
129, 10, 11sylancr 694 . 2 (𝐵 ∈ On → ( (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
137, 12syl5bb 271 1 (𝐵 ∈ On → (∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372  ∪ ciun 4455  Ord word 5639  Oncon0 5640  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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