Theorem rankbnd2 9795

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 9789	. . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)
2		rankr1b.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	rankuni2 9781	. . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
4	1, 3	eqtr3i 2762	. . . 4 ⊢ ∪ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
5	4	sseq1i 3964	. . 3 ⊢ (∪ (rank‘𝐴) ⊆ 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵)
6		iunss 5002	. . 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 9721	. . . 4 ⊢ (rank‘𝐴) ∈ On
9	8	onssi 7792	. . 3 ⊢ (rank‘𝐴) ⊆ On
10		eloni 6337	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
11		ordunisssuc 6435	. . 3 ⊢ (((rank‘𝐴) ⊆ On ∧ Ord 𝐵) → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))
12	9, 10, 11	sylancr 588	. 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 2114  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  ∪ cuni 4865  ∪ ciun 4948  Ord word 6326  Oncon0 6327  suc csuc 6329  ‘cfv 6502  rankcrnk 9689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-reg 9511  ax-inf2 9564

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-r1 9690  df-rank 9691

This theorem is referenced by: (None)