Theorem harcard 9900

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 9471	. 2 ⊢ (har‘𝐴) ∈ On
2		harndom 9474	. . . . . . 7 ⊢ ¬ (har‘𝐴) ≼ 𝐴
3		simpll 772	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4		elharval 9473	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
5	4	bilani 505	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
6	5	simpld 495	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
7		ontri1 6351	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
8	3, 6, 7	syl2anc 590	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
9		simpllr 781	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≈ 𝑥)
10		ssdomg 8944	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦))
11	10	elv 3437	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦)
12	5	simprd 496	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ≼ 𝐴)
13		domtr 8951	. . . . . . . . . . 11 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝐴) → 𝑥 ≼ 𝐴)
14	11, 12, 13	syl2anr 603	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ≼ 𝐴)
15		endomtr 8956	. . . . . . . . . 10 ⊢ (((har‘𝐴) ≈ 𝑥 ∧ 𝑥 ≼ 𝐴) → (har‘𝐴) ≼ 𝐴)
16	9, 14, 15	syl2anc 590	. . . . . . . . 9 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≼ 𝐴)
17	16	ex 413	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥 ⊆ 𝑦 → (har‘𝐴) ≼ 𝐴))
18	8, 17	sylbird 261	. . . . . . 7 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (¬ 𝑦 ∈ 𝑥 → (har‘𝐴) ≼ 𝐴))
19	2, 18	mt3i 149	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ 𝑥)
20	19	ex 413	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦 ∈ 𝑥))
21	20	ssrdv 3928	. . . 4 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
22	21	ex 413	. . 3 ⊢ (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
23	22	rgen 3056	. 2 ⊢ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
24		iscard2 9898	. 2 ⊢ ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
25	1, 23, 24	mpbir2an 717	1 ⊢ (card‘(har‘𝐴)) = (har‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   ⊆ wss 3890   class class class wbr 5079  Oncon0 6317  ‘cfv 6492   ≈ cen 8887   ≼ cdom 8888  harchar 9468  cardccrd 9857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-er 8640  df-en 8891  df-dom 8892  df-oi 9422  df-har 9469  df-card 9861

This theorem is referenced by:  cardprclem  9901  alephcard  9990  pwcfsdom  10504  hargch  10594