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Theorem rankss 9070

Description: The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

**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	. . 3 ⊢ 𝐵 ∈ V
2		unir1 9034	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2858	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
4		rankssb 9069	. 2 ⊢ (𝐵 ∈ ∪ (𝑅₁ “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))
5	3, 4	ax-mp 5	1 ⊢ (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2051 Vcvv 3408 ⊆ wss 3822 ∪ cuni 4708 “ cima 5406 Oncon0 6026 ‘cfv 6185 𝑅₁cr1 8983 rankcrnk 8984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-reg 8849 ax-inf2 8896

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-r1 8985 df-rank 8986

This theorem is referenced by: rankuni 9084 rankval4 9088 rankc2 9092 rankxpu 9097 rankfu 9098 rankmapu 9099 rankxplim 9100

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