Theorem rankbnd2 9627

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 9621	. . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)
2		rankr1b.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	rankuni2 9613	. . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
4	1, 3	eqtr3i 2768	. . . 4 ⊢ ∪ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
5	4	sseq1i 3949	. . 3 ⊢ (∪ (rank‘𝐴) ⊆ 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵)
6		iunss 4975	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵)
7	5, 6	bitr2i 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∪ (rank‘𝐴) ⊆ 𝐵)
8		rankon 9553	. . . 4 ⊢ (rank‘𝐴) ∈ On
9	8	onssi 7684	. . 3 ⊢ (rank‘𝐴) ⊆ On
10		eloni 6276	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
11		ordunisssuc 6368	. . 3 ⊢ (((rank‘𝐴) ⊆ On ∧ Ord 𝐵) → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
12	9, 10, 11	sylancr 587	. 2 ⊢ (𝐵 ∈ On → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
13	7, 12	bitrid 282	1 ⊢ (𝐵 ∈ On → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839  ∪ ciun 4924  Ord word 6265  Oncon0 6266  suc csuc 6268  ‘cfv 6433  rankcrnk 9521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523

This theorem is referenced by: (None)