Theorem harcard 9973

Description: The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
harcard	⊢ (card‘(har‘𝐴)) = (har‘𝐴)

Proof of Theorem harcard

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		harcl 9554	. 2 ⊢ (har‘𝐴) ∈ On
2		harndom 9557	. . . . . . 7 ⊢ ¬ (har‘𝐴) ≼ 𝐴
3		simpll 766	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4		simpr 486	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ (har‘𝐴))
5		elharval 9556	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
6	4, 5	sylib 217	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
7	6	simpld 496	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
8		ontri1 6399	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
9	3, 7, 8	syl2anc 585	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
10		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≈ 𝑥)
11		ssdomg 8996	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦))
12	11	elv 3481	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦)
13	6	simprd 497	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ≼ 𝐴)
14		domtr 9003	. . . . . . . . . . 11 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝐴) → 𝑥 ≼ 𝐴)
15	12, 13, 14	syl2anr 598	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ≼ 𝐴)
16		endomtr 9008	. . . . . . . . . 10 ⊢ (((har‘𝐴) ≈ 𝑥 ∧ 𝑥 ≼ 𝐴) → (har‘𝐴) ≼ 𝐴)
17	10, 15, 16	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≼ 𝐴)
18	17	ex 414	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥 ⊆ 𝑦 → (har‘𝐴) ≼ 𝐴))
19	9, 18	sylbird 260	. . . . . . 7 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (¬ 𝑦 ∈ 𝑥 → (har‘𝐴) ≼ 𝐴))
20	2, 19	mt3i 149	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ 𝑥)
21	20	ex 414	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦 ∈ 𝑥))
22	21	ssrdv 3989	. . . 4 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
23	22	ex 414	. . 3 ⊢ (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
24	23	rgen 3064	. 2 ⊢ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
25		iscard2 9971	. 2 ⊢ ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
26	1, 24, 25	mpbir2an 710	1 ⊢ (card‘(har‘𝐴)) = (har‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ⊆ wss 3949   class class class wbr 5149  Oncon0 6365  ‘cfv 6544   ≈ cen 8936   ≼ cdom 8937  harchar 9551  cardccrd 9930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-er 8703  df-en 8940  df-dom 8941  df-oi 9505  df-har 9552  df-card 9934

This theorem is referenced by:  cardprclem  9974  alephcard  10065  pwcfsdom  10578  hargch  10668