Theorem harcard 9893

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 9470	. 2 ⊢ (har‘𝐴) ∈ On
2		harndom 9473	. . . . . . 7 ⊢ ¬ (har‘𝐴) ≼ 𝐴
3		simpll 766	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ (har‘𝐴))
5		elharval 9472	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
6	4, 5	sylib 218	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
7	6	simpld 494	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
8		ontri1 6345	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
9	3, 7, 8	syl2anc 584	. . . . . . . 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 8932	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦))
12	11	elv 3443	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦)
13	6	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ≼ 𝐴)
14		domtr 8939	. . . . . . . . . . 11 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝐴) → 𝑥 ≼ 𝐴)
15	12, 13, 14	syl2anr 597	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ≼ 𝐴)
16		endomtr 8944	. . . . . . . . . 10 ⊢ (((har‘𝐴) ≈ 𝑥 ∧ 𝑥 ≼ 𝐴) → (har‘𝐴) ≼ 𝐴)
17	10, 15, 16	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≼ 𝐴)
18	17	ex 412	. . . . . . . 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 412	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦 ∈ 𝑥))
22	21	ssrdv 3943	. . . 4 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
23	22	ex 412	. . 3 ⊢ (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
24	23	rgen 3046	. 2 ⊢ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
25		iscard2 9891	. 2 ⊢ ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
26	1, 24, 25	mpbir2an 711	1 ⊢ (card‘(har‘𝐴)) = (har‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438   ⊆ wss 3905   class class class wbr 5095  Oncon0 6311  ‘cfv 6486   ≈ cen 8876   ≼ cdom 8877  harchar 9467  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-er 8632  df-en 8880  df-dom 8881  df-oi 9421  df-har 9468  df-card 9854

This theorem is referenced by:  cardprclem  9894  alephcard  9983  pwcfsdom  10496  hargch  10586