Theorem rankr1lem 4683

Description: Lemma for rankr1 4684.

Hypothesis

Ref	Expression
rankr1lem.1	⊢ A ∈ V

Assertion

Ref	Expression
rankr1lem	⊢ (B ∈ On → (¬ A ∈ (R1 ‘B) → B ⊆ (rank ‘A)))

Proof of Theorem rankr1lem

Step	Hyp	Ref	Expression
1		rankon 4681	. . . . . 6 ⊢ (rank ‘A) ∈ On
2	1	onsuc 3111	. . . . 5 ⊢ suc (rank ‘A) ∈ On
3		ontri1 2987	. . . . 5 ⊢ ((suc (rank ‘A) ∈ On ⋀ B ∈ On) → (suc (rank ‘A) ⊆ B ↔ ¬ B ∈ suc (rank ‘A)))
4	2, 3	mpan 697	. . . 4 ⊢ (B ∈ On → (suc (rank ‘A) ⊆ B ↔ ¬ B ∈ suc (rank ‘A)))
5		rankr1lem.1	. . . . . 6 ⊢ A ∈ V
6	5	rankid 4682	. . . . 5 ⊢ A ∈ (R1 ‘suc (rank ‘A))
7		r1ord3 4667	. . . . . . 7 ⊢ ((suc (rank ‘A) ∈ On ⋀ B ∈ On) → (suc (rank ‘A) ⊆ B → (R1 ‘suc (rank ‘A)) ⊆ (R1 ‘B)))
8	2, 7	mpan 697	. . . . . 6 ⊢ (B ∈ On → (suc (rank ‘A) ⊆ B → (R1 ‘suc (rank ‘A)) ⊆ (R1 ‘B)))
9		ssel 2066	. . . . . 6 ⊢ ((R1 ‘suc (rank ‘A)) ⊆ (R1 ‘B) → (A ∈ (R1 ‘suc (rank ‘A)) → A ∈ (R1 ‘B)))
10	8, 9	syl6 22	. . . . 5 ⊢ (B ∈ On → (suc (rank ‘A) ⊆ B → (A ∈ (R1 ‘suc (rank ‘A)) → A ∈ (R1 ‘B))))
11	6, 10	mpii 45	. . . 4 ⊢ (B ∈ On → (suc (rank ‘A) ⊆ B → A ∈ (R1 ‘B)))
12	4, 11	sylbird 205	. . 3 ⊢ (B ∈ On → (¬ B ∈ suc (rank ‘A) → A ∈ (R1 ‘B)))
13	12	con1d 93	. 2 ⊢ (B ∈ On → (¬ A ∈ (R1 ‘B) → B ∈ suc (rank ‘A)))
14		onsssuc 3064	. . 3 ⊢ ((B ∈ On ⋀ (rank ‘A) ∈ On) → (B ⊆ (rank ‘A) ↔ B ∈ suc (rank ‘A)))
15	1, 14	mpan2 698	. 2 ⊢ (B ∈ On → (B ⊆ (rank ‘A) ↔ B ∈ suc (rank ‘A)))
16	13, 15	sylibrd 204	1 ⊢ (B ∈ On → (¬ A ∈ (R1 ‘B) → B ⊆ (rank ‘A)))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ∈ wcel 960  Vcvv 1814   ⊆ wss 2050  Oncon0 2954  suc csuc 2956   ‘cfv 3188  R₁cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankr1 4684  ssrankr1 4686

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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