Theorem isinfcard 4898

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

Assertion

Ref	Expression
isinfcard	|- ((om (_ A /\ (card` A) = A) <-> A e. ran aleph)

Proof of Theorem isinfcard

Step	Hyp	Ref	Expression
1		eqcom 1480	. . 3 |- ((aleph` x) = A <-> A = (aleph` x))
2	1	rexbii 1671	. 2 |- (E.x e. On (aleph` x) = A <-> E.x e. On A = (aleph` x))
3		alephfnon 4873	. . 3 |- aleph Fn On
4		fvelrnb 3766	. . 3 |- (aleph Fn On -> (A e. ran aleph <-> E.x e. On (aleph` x) = A))
5	3, 4	ax-mp 7	. 2 |- (A e. ran aleph <-> E.x e. On (aleph` x) = A)
6		cardalephex 4897	. . . 4 |- (om (_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))
7	6	pm5.32i 647	. . 3 |- ((om (_ A /\ (card` A) = A) <-> (om (_ A /\ E.x e. On A = (aleph` x)))
8		sseq2 2086	. . . . . 6 |- (A = (aleph` x) -> (om (_ A <-> om (_ (aleph` x)))
9		alephgeom 4893	. . . . . . 7 |- (x e. On <-> om (_ (aleph` x))
10	9	biimp 151	. . . . . 6 |- (x e. On -> om (_ (aleph` x))
11	8, 10	syl5cbir 211	. . . . 5 |- (x e. On -> (A = (aleph` x) -> om (_ A))
12	11	r19.23aiv 1746	. . . 4 |- (E.x e. On A = (aleph` x) -> om (_ A)
13	12	pm4.71ri 640	. . 3 |- (E.x e. On A = (aleph` x) <-> (om (_ A /\ E.x e. On A = (aleph` x)))
14	7, 13	bitr4 176	. 2 |- ((om (_ A /\ (card` A) = A) <-> E.x e. On A = (aleph` x))
15	2, 5, 14	3bitr4r 184	1 |- ((om (_ A /\ (card` A) = A) <-> A e. ran aleph)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wrex 1649   (_ wss 2050  Oncon0 2954  omcom 3137  ran crn 3177   Fn wfn 3183  ` cfv 3188  cardccrd 4823  alephcale 4824

This theorem is referenced by:  iscard3 4899  cardinfima 4902  alephiso 4903  alephsson 4905  alephfp 4911

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  ax-ac 4754

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-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  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-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376  df-fin 4377  df-card 4826  df-aleph 4827

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