Theorem iscard 4866

Description: Two ways to express the property of being a cardinal number.

Assertion

Ref	Expression
iscard	⊢ ((card ‘A) = A ↔ (A ∈ On ⋀ ∀x ∈ A x ≺ A))

Distinct variable group:   x,A

Proof of Theorem iscard

Step	Hyp	Ref	Expression
1		cardon 4839	. . . 4 ⊢ (card ‘A) ∈ On
2		eleq1 1541	. . . 4 ⊢ ((card ‘A) = A → ((card ‘A) ∈ On ↔ A ∈ On))
3	1, 2	mpbii 193	. . 3 ⊢ ((card ‘A) = A → A ∈ On)
4	3	pm4.71ri 641	. 2 ⊢ ((card ‘A) = A ↔ (A ∈ On ⋀ (card ‘A) = A))
5		cardonle 4834	. . . . 5 ⊢ (A ∈ On → (card ‘A) ⊆ A)
6		eqss 2086	. . . . . 6 ⊢ ((card ‘A) = A ↔ ((card ‘A) ⊆ A ⋀ A ⊆ (card ‘A)))
7	6	baibr 690	. . . . 5 ⊢ ((card ‘A) ⊆ A → (A ⊆ (card ‘A) ↔ (card ‘A) = A))
8	5, 7	syl 10	. . . 4 ⊢ (A ∈ On → (A ⊆ (card ‘A) ↔ (card ‘A) = A))
9		onelon 2986	. . . . . . 7 ⊢ ((A ∈ On ⋀ x ∈ A) → x ∈ On)
10		cardsdomel 4865	. . . . . . 7 ⊢ (x ∈ On → (x ≺ A ↔ x ∈ (card ‘A)))
11	9, 10	syl 10	. . . . . 6 ⊢ ((A ∈ On ⋀ x ∈ A) → (x ≺ A ↔ x ∈ (card ‘A)))
12	11	ralbidva 1666	. . . . 5 ⊢ (A ∈ On → (∀x ∈ A x ≺ A ↔ ∀x ∈ A x ∈ (card ‘A)))
13		dfss3 2068	. . . . 5 ⊢ (A ⊆ (card ‘A) ↔ ∀x ∈ A x ∈ (card ‘A))
14	12, 13	syl6rbbr 542	. . . 4 ⊢ (A ∈ On → (A ⊆ (card ‘A) ↔ ∀x ∈ A x ≺ A))
15	8, 14	bitr3d 533	. . 3 ⊢ (A ∈ On → ((card ‘A) = A ↔ ∀x ∈ A x ≺ A))
16	15	pm5.32i 648	. 2 ⊢ ((A ∈ On ⋀ (card ‘A) = A) ↔ (A ∈ On ⋀ ∀x ∈ A x ≺ A))
17	4, 16	bitr 173	1 ⊢ ((card ‘A) = A ↔ (A ∈ On ⋀ ∀x ∈ A x ≺ A))

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 960   ∈ wcel 962  ∀wral 1652   ⊆ wss 2056   class class class wbr 2632  Oncon0 2962   ‘cfv 3196   ≺ csdm 4380  cardccrd 4825

This theorem is referenced by:  cardmin 4873

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880  ax-ac 4756

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 780  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-sbc 1949  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-tp 2425  df-op 2426  df-uni 2516  df-int 2546  df-iun 2580  df-br 2633  df-opab 2680  df-tr 2694  df-eprel 2846  df-id 2849  df-po 2854  df-so 2864  df-fr 2931  df-we 2948  df-ord 2965  df-on 2966  df-suc 2968  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-f1 3209  df-fo 3210  df-f1o 3211  df-fv 3212  df-er 4275  df-en 4382  df-dom 4383  df-sdom 4384  df-card 4828

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