rankss - Metamath Proof Explorer

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Theorem rankss 9846

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 9810	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2832	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
4		rankssb 9845	. 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 2106 Vcvv 3474 ⊆ wss 3948 ∪ cuni 4908 “ cima 5679 Oncon0 6364 ‘cfv 6543 𝑅₁cr1 9759 rankcrnk 9760

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-reg 9589 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-r1 9761 df-rank 9762

This theorem is referenced by: rankuni 9860 rankval4 9864 rankc2 9868 rankxpu 9873 rankfu 9874 rankmapu 9875 rankxplim 9876

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