Theorem rankc1 4715

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

Hypothesis

Ref	Expression
rankr1b.1	⊢ A ∈ V

Assertion

Ref	Expression
rankc1	⊢ (∀x ∈ A (rank ‘x) ∈ (rank ‘∪A) ↔ (rank ‘A) = (rank ‘∪A))

Distinct variable group:   x,A

Proof of Theorem rankc1

Step	Hyp	Ref	Expression
1		rankr1b.1	. . . 4 ⊢ A ∈ V
2	1	rankuniss 4711	. . 3 ⊢ (rank ‘∪A) ⊆ (rank ‘A)
3	2	biantru 726	. 2 ⊢ ((rank ‘A) ⊆ (rank ‘∪A) ↔ ((rank ‘A) ⊆ (rank ‘∪A) ⋀ (rank ‘∪A) ⊆ (rank ‘A)))
4		iunss 2595	. . 3 ⊢ (∪x ∈ A suc (rank ‘x) ⊆ (rank ‘∪A) ↔ ∀x ∈ A suc (rank ‘x) ⊆ (rank ‘∪A))
5	1	rankval4 4712	. . . 4 ⊢ (rank ‘A) = ∪x ∈ A suc (rank ‘x)
6	5	sseq1i 2088	. . 3 ⊢ ((rank ‘A) ⊆ (rank ‘∪A) ↔ ∪x ∈ A suc (rank ‘x) ⊆ (rank ‘∪A))
7		rankon 4681	. . . . 5 ⊢ (rank ‘x) ∈ On
8		rankon 4681	. . . . 5 ⊢ (rank ‘∪A) ∈ On
9	7, 8	onsucss 3117	. . . 4 ⊢ ((rank ‘x) ∈ (rank ‘∪A) ↔ suc (rank ‘x) ⊆ (rank ‘∪A))
10	9	ralbii 1670	. . 3 ⊢ (∀x ∈ A (rank ‘x) ∈ (rank ‘∪A) ↔ ∀x ∈ A suc (rank ‘x) ⊆ (rank ‘∪A))
11	4, 6, 10	3bitr4r 184	. 2 ⊢ (∀x ∈ A (rank ‘x) ∈ (rank ‘∪A) ↔ (rank ‘A) ⊆ (rank ‘∪A))
12		eqss 2080	. 2 ⊢ ((rank ‘A) = (rank ‘∪A) ↔ ((rank ‘A) ⊆ (rank ‘∪A) ⋀ (rank ‘∪A) ⊆ (rank ‘A)))
13	3, 11, 12	3bitr4 183	1 ⊢ (∀x ∈ A (rank ‘x) ∈ (rank ‘∪A) ↔ (rank ‘A) = (rank ‘∪A))

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648  Vcvv 1814   ⊆ wss 2050  ∪cuni 2507  ∪ciun 2570  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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