Theorem rankss 3531

Description: The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80.

Hypothesis

Ref	Expression
rankss.1	⊢ B ∈ V

Assertion

Ref	Expression
rankss	⊢ (A ⊆ B → (rank ‘A) ⊆ (rank ‘B))

Proof of Theorem rankss

Step	Hyp	Ref	Expression
1		rankss.1	. . . 4 ⊢ B ∈ V
2	1	pwex 1806	. . 3 ⊢ ℘B ∈ V
3	2	rankel 3524	. 2 ⊢ (A ∈ ℘B → (rank ‘A) ∈ (rank ‘℘B))
4		elpw2g 1803	. . 3 ⊢ (B ∈ V → (A ∈ ℘B ↔ A ⊆ B))
5	1, 4	ax-mp 6	. 2 ⊢ (A ∈ ℘B ↔ A ⊆ B)
6	1	rankpw 3528	. . . 4 ⊢ (rank ‘℘B) = suc (rank ‘B)
7	6	eleq2i 1153	. . 3 ⊢ ((rank ‘A) ∈ (rank ‘℘B) ↔ (rank ‘A) ∈ suc (rank ‘B))
8		rankon 3515	. . . 4 ⊢ (rank ‘A) ∈ On
9		rankon 3515	. . . 4 ⊢ (rank ‘B) ∈ On
10		onsssuc 2311	. . . 4 ⊢ (((rank ‘A) ∈ On ∧ (rank ‘B) ∈ On) → ((rank ‘A) ⊆ (rank ‘B) ↔ (rank ‘A) ∈ suc (rank ‘B)))
11	8, 9, 10	mp2an 520	. . 3 ⊢ ((rank ‘A) ⊆ (rank ‘B) ↔ (rank ‘A) ∈ suc (rank ‘B))
12	7, 11	bitr4 154	. 2 ⊢ ((rank ‘A) ∈ (rank ‘℘B) ↔ (rank ‘A) ⊆ (rank ‘B))
13	3, 5, 12	3imtr3 191	1 ⊢ (A ⊆ B → (rank ‘A) ⊆ (rank ‘B))

Syntax hints:   → wi 2   ↔ wb 127   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  Oncon0 2199  suc csuc 2201   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  rankuni 3533  rankun 3535  rankr1id 3539  ranklon 3540

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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