Theorem rankc2 4716

Description: A relationship that can be used for computation of rank.

Hypothesis

Ref	Expression
rankr1b.1	⊢ A ∈ V

Assertion

Ref	Expression
rankc2	⊢ (∃x ∈ A (rank ‘x) = (rank ‘∪A) → (rank ‘A) = suc (rank ‘∪A))

Distinct variable group:   x,A

Proof of Theorem rankc2

Step	Hyp	Ref	Expression
1		pwuni 2763	. . . . 5 ⊢ A ⊆ ℘∪A
2		rankr1b.1	. . . . . . . 8 ⊢ A ∈ V
3	2	uniex 2876	. . . . . . 7 ⊢ ∪A ∈ V
4	3	pwex 2751	. . . . . 6 ⊢ ℘∪A ∈ V
5	4	rankss 4698	. . . . 5 ⊢ (A ⊆ ℘∪A → (rank ‘A) ⊆ (rank ‘℘∪A))
6	1, 5	ax-mp 7	. . . 4 ⊢ (rank ‘A) ⊆ (rank ‘℘∪A)
7	3	rankpw 4694	. . . 4 ⊢ (rank ‘℘∪A) = suc (rank ‘∪A)
8	6, 7	sseqtr 2096	. . 3 ⊢ (rank ‘A) ⊆ suc (rank ‘∪A)
9	8	a1i 8	. 2 ⊢ (∃x ∈ A (rank ‘x) = (rank ‘∪A) → (rank ‘A) ⊆ suc (rank ‘∪A))
10		eleq1 1537	. . . . 5 ⊢ ((rank ‘x) = (rank ‘∪A) → ((rank ‘x) ∈ (rank ‘A) ↔ (rank ‘∪A) ∈ (rank ‘A)))
11	2	rankel 4690	. . . . 5 ⊢ (x ∈ A → (rank ‘x) ∈ (rank ‘A))
12	10, 11	syl5cbi 209	. . . 4 ⊢ (x ∈ A → ((rank ‘x) = (rank ‘∪A) → (rank ‘∪A) ∈ (rank ‘A)))
13	12	r19.23aiv 1746	. . 3 ⊢ (∃x ∈ A (rank ‘x) = (rank ‘∪A) → (rank ‘∪A) ∈ (rank ‘A))
14		rankon 4681	. . . 4 ⊢ (rank ‘∪A) ∈ On
15		rankon 4681	. . . 4 ⊢ (rank ‘A) ∈ On
16	14, 15	onsucss 3117	. . 3 ⊢ ((rank ‘∪A) ∈ (rank ‘A) ↔ suc (rank ‘∪A) ⊆ (rank ‘A))
17	13, 16	sylib 198	. 2 ⊢ (∃x ∈ A (rank ‘x) = (rank ‘∪A) → suc (rank ‘∪A) ⊆ (rank ‘A))
18	9, 17	eqssd 2082	1 ⊢ (∃x ∈ A (rank ‘x) = (rank ‘∪A) → (rank ‘A) = suc (rank ‘∪A))

Syntax hints:   → wi 3   = wceq 958   ∈ wcel 960  ∃wrex 1649  Vcvv 1814   ⊆ wss 2050  ℘cpw 2405  ∪cuni 2507  suc csuc 2956   ‘cfv 3188  rankcrnk 4652

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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