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Theorem rankr1 9338

Description: A relationship between the rank function and the cumulative hierarchy of sets function 𝑅₁. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)

**Hypothesis**
Ref	Expression
rankid.1	⊢ 𝐴 ∈ V

**Assertion**
Ref	Expression
rankr1	⊢ (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅₁‘𝐵) ∧ 𝐴 ∈ (𝑅₁‘suc 𝐵)))

**Proof of Theorem rankr1**
Step	Hyp	Ref	Expression
1		rankid.1	. 2 ⊢ 𝐴 ∈ V
2		rankr1g 9336	. 2 ⊢ (𝐴 ∈ V → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅₁‘𝐵) ∧ 𝐴 ∈ (𝑅₁‘suc 𝐵))))
3	1, 2	ax-mp 5	1 ⊢ (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅₁‘𝐵) ∧ 𝐴 ∈ (𝑅₁‘suc 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 suc csuc 6174 ‘cfv 6339 𝑅₁cr1 9266 rankcrnk 9267

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 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-reg 9131 ax-inf2 9179

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7602 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-r1 9268 df-rank 9269

This theorem is referenced by: (None)

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