Theorem rankbnd2 9938

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 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rankbnd2

Step	Hyp	Ref	Expression
1		rankuni 9932	. . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)
2		rankr1b.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	rankuni2 9924	. . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
4	1, 3	eqtr3i 2770	. . . 4 ⊢ ∪ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
5	4	sseq1i 4037	. . 3 ⊢ (∪ (rank‘𝐴) ⊆ 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵)
6		iunss 5068	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵)
7	5, 6	bitr2i 276	. 2 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∪ (rank‘𝐴) ⊆ 𝐵)
8		rankon 9864	. . . 4 ⊢ (rank‘𝐴) ∈ On
9	8	onssi 7874	. . 3 ⊢ (rank‘𝐴) ⊆ On
10		eloni 6405	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
11		ordunisssuc 6501	. . 3 ⊢ (((rank‘𝐴) ⊆ On ∧ Ord 𝐵) → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
12	9, 10, 11	sylancr 586	. 2 ⊢ (𝐵 ∈ On → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
13	7, 12	bitrid 283	1 ⊢ (𝐵 ∈ On → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∪ cuni 4931  ∪ ciun 5015  Ord word 6394  Oncon0 6395  suc csuc 6397  ‘cfv 6573  rankcrnk 9832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-r1 9833  df-rank 9834

This theorem is referenced by: (None)