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Theorem rankss 9831

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 9795	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2833	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
4		rankssb 9830	. 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 2107 Vcvv 3475 ⊆ wss 3946 ∪ cuni 4904 “ cima 5675 Oncon0 6356 ‘cfv 6535 𝑅₁cr1 9744 rankcrnk 9745

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-reg 9574 ax-inf2 9623

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-r1 9746 df-rank 9747

This theorem is referenced by: rankuni 9845 rankval4 9849 rankc2 9853 rankxpu 9858 rankfu 9859 rankmapu 9860 rankxplim 9861

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