rankel - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rankel		Structured version Visualization version GIF version

Theorem rankel 9267

Description: The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

**Hypothesis**
Ref	Expression
rankel.1	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
rankel	⊢ (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵))

**Proof of Theorem rankel**
Step	Hyp	Ref	Expression
1		rankel.1	. . 3 ⊢ 𝐵 ∈ V
2		unir1 9241	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2912	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
4		rankelb 9252	. 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 2110 Vcvv 3494 ∪ cuni 4837 “ cima 5557 Oncon0 6190 ‘cfv 6354 𝑅₁cr1 9190 rankcrnk 9191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-reg 9055 ax-inf2 9103

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-r1 9192 df-rank 9193

This theorem is referenced by: rankuni 9291 rankval4 9295 rankc2 9299 rankxplim 9307 rankelg 33629

W3C validator