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Theorem rankop 8885

Description: The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

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

**Assertion**
Ref	Expression
rankop	⊢ (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))

**Proof of Theorem rankop**
Step	Hyp	Ref	Expression
1		ranksn.1	. . 3 ⊢ 𝐴 ∈ V
2		unir1 8840	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2849	. 2 ⊢ 𝐴 ∈ ∪ (𝑅₁ “ On)
4		rankun.2	. . 3 ⊢ 𝐵 ∈ V
5	4, 2	eleqtrri 2849	. 2 ⊢ 𝐵 ∈ ∪ (𝑅₁ “ On)
6		rankopb 8879	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝐵 ∈ ∪ (𝑅₁ “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
7	3, 5, 6	mp2an 672	1 ⊢ (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∪ cun 3721 ⟨cop 4322 ∪ cuni 4574 “ cima 5252 Oncon0 5866 suc csuc 5868 ‘cfv 6031 𝑅₁cr1 8789 rankcrnk 8790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-reg 8653 ax-inf2 8702

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-r1 8791 df-rank 8792

This theorem is referenced by: rankelop 8901

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