Theorem ssrankr1 3520

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.

Hypothesis

Ref	Expression
ssrankr1.1	⊢ A ∈ V

Assertion

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

Proof of Theorem ssrankr1

Step	Hyp	Ref	Expression
1		cleqid 1102	. . . . . 6 ⊢ (rank ‘A) = (rank ‘A)
2		ssrankr1.1	. . . . . . 7 ⊢ A ∈ V
3	2	rankr1 3518	. . . . . 6 ⊢ ((rank ‘A) = (rank ‘A) ↔ (¬ A ∈ (R1 ‘(rank ‘A)) ∧ A ∈ (R1 ‘suc (rank ‘A))))
4	1, 3	mpbi 164	. . . . 5 ⊢ (¬ A ∈ (R1 ‘(rank ‘A)) ∧ A ∈ (R1 ‘suc (rank ‘A)))
5	4	pm3.26i 257	. . . 4 ⊢ ¬ A ∈ (R1 ‘(rank ‘A))
6		rankon 3515	. . . . . . 7 ⊢ (rank ‘A) ∈ On
7		r1ord3 3501	. . . . . . 7 ⊢ ((B ∈ On ∧ (rank ‘A) ∈ On) → (B ⊆ (rank ‘A) → (R1 ‘B) ⊆ (R1 ‘(rank ‘A))))
8	6, 7	mpan2 519	. . . . . 6 ⊢ (B ∈ On → (B ⊆ (rank ‘A) → (R1 ‘B) ⊆ (R1 ‘(rank ‘A))))
9	8	imp 277	. . . . 5 ⊢ ((B ∈ On ∧ B ⊆ (rank ‘A)) → (R1 ‘B) ⊆ (R1 ‘(rank ‘A)))
10	9	sseld 1506	. . . 4 ⊢ ((B ∈ On ∧ B ⊆ (rank ‘A)) → (A ∈ (R1 ‘B) → A ∈ (R1 ‘(rank ‘A))))
11	5, 10	mtoi 94	. . 3 ⊢ ((B ∈ On ∧ B ⊆ (rank ‘A)) → ¬ A ∈ (R1 ‘B))
12	11	exp 291	. 2 ⊢ (B ∈ On → (B ⊆ (rank ‘A) → ¬ A ∈ (R1 ‘B)))
13	2	rankr1lem 3517	. 2 ⊢ (B ∈ On → (¬ A ∈ (R1 ‘B) → B ⊆ (rank ‘A)))
14	12, 13	impbid 397	1 ⊢ (B ∈ On → (B ⊆ (rank ‘A) ↔ ¬ A ∈ (R1 ‘B)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  Oncon0 2199  suc csuc 2201   ‘cfv 2422  R₁cr1 3485  rankcrnk 3486

This theorem is referenced by:  rankr1a 3521  r1val2 3522

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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