rankun - Metamath Proof Explorer

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

Theorem rankun 9865

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 9822	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2827	. 2 ⊢ 𝐴 ∈ ∪ (𝑅₁ “ On)
4		rankun.2	. . 3 ⊢ 𝐵 ∈ V
5	4, 2	eleqtrri 2827	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
6		rankunb 9859	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝐵 ∈ ∪ (𝑅₁ “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
7	3, 5, 6	mp2an 691	1 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cun 3942 ∪ cuni 4903 “ cima 5675 Oncon0 6363 ‘cfv 6542 𝑅₁cr1 9771 rankcrnk 9772

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-reg 9601 ax-inf2 9650

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-r1 9773 df-rank 9774

This theorem is referenced by: ranksuc 9874 rankelun 9881 rankelpr 9882 rankung 35685

W3C validator