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Theorem rankun 9772

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 9729	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2838	. 2 ⊢ 𝐴 ∈ ∪ (𝑅₁ “ On)
4		rankun.2	. . 3 ⊢ 𝐵 ∈ V
5	4, 2	eleqtrri 2838	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
6		rankunb 9766	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝐵 ∈ ∪ (𝑅₁ “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
7	3, 5, 6	mp2an 698	1 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 ∈ wcel 2119 Vcvv 3431 ∪ cun 3881 ∪ cuni 4839 “ cima 5622 Oncon0 6311 ‘cfv 6486 𝑅₁cr1 9678 rankcrnk 9679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-reg 9498 ax-inf2 9554

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7360 df-om 7808 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-r1 9680 df-rank 9681

This theorem is referenced by: ranksuc 9781 rankelun 9788 rankelpr 9789 rankung 36403

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