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Theorem rankun 9749

Description: The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by NM, 26-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

**Hypotheses**
Ref	Expression
ranksn.1	⊢ 𝐴 ∈ V
rankun.2	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
rankun	⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))

**Proof of Theorem rankun**
Step	Hyp	Ref	Expression
1		ranksn.1	. . 3 ⊢ 𝐴 ∈ V
2		unir1 9706	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2830	. 2 ⊢ 𝐴 ∈ ∪ (𝑅₁ “ On)
4		rankun.2	. . 3 ⊢ 𝐵 ∈ V
5	4, 2	eleqtrri 2830	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
6		rankunb 9743	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝐵 ∈ ∪ (𝑅₁ “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
7	3, 5, 6	mp2an 692	1 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 Vcvv 3436 ∪ cun 3895 ∪ cuni 4856 “ cima 5617 Oncon0 6306 ‘cfv 6481 𝑅₁cr1 9655 rankcrnk 9656

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-reg 9478 ax-inf2 9531

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-r1 9657 df-rank 9658

This theorem is referenced by: ranksuc 9758 rankelun 9765 rankelpr 9766 rankung 36210

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