Theorem ssrankr1 8871

Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets 𝑅₁. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Hypothesis

Ref	Expression
rankid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ssrankr1	⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1‘𝐵)))

Proof of Theorem ssrankr1

Step	Hyp	Ref	Expression
1		rankid.1	. . . 4 ⊢ 𝐴 ∈ V
2		unir1 8849	. . . 4 ⊢ ∪ (𝑅1 “ On) = V
3	1, 2	eleqtrri 2838	. . 3 ⊢ 𝐴 ∈ ∪ (𝑅1 “ On)
4		r1fnon 8803	. . . . . 6 ⊢ 𝑅1 Fn On
5		fndm 6151	. . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
6	4, 5	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 = On
7	6	eleq2i 2831	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On)
8	7	biimpri 218	. . 3 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom 𝑅1)
9		rankr1clem 8856	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
10	3, 8, 9	sylancr 698	. 2 ⊢ (𝐵 ∈ On → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
11	10	bicomd 213	1 ⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ⊆ wss 3715  ∪ cuni 4588  dom cdm 5266   “ cima 5269  Oncon0 5884   Fn wfn 6044  ‘cfv 6049  𝑅₁cr1 8798  rankcrnk 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-reg 8662  ax-inf2 8711

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-rank 8801

This theorem is referenced by:  rankr1a  8872