Theorem harcard 9933

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 9504	. 2 ⊢ (har‘𝐴) ∈ On
2		harndom 9507	. . . . . . 7 ⊢ ¬ (har‘𝐴) ≼ 𝐴
3		simpll 776	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4		elharval 9506	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
5	4	bilani 508	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
6	5	simpld 498	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
7		ontri1 6376	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
8	3, 6, 7	syl2anc 593	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
9		simpllr 785	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≈ 𝑥)
10		ssdomg 8977	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦))
11	10	elv 3458	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦)
12	5	simprd 499	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ≼ 𝐴)
13		domtr 8984	. . . . . . . . . . 11 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝐴) → 𝑥 ≼ 𝐴)
14	11, 12, 13	syl2anr 606	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ≼ 𝐴)
15		endomtr 8989	. . . . . . . . . 10 ⊢ (((har‘𝐴) ≈ 𝑥 ∧ 𝑥 ≼ 𝐴) → (har‘𝐴) ≼ 𝐴)
16	9, 14, 15	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≼ 𝐴)
17	16	ex 416	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥 ⊆ 𝑦 → (har‘𝐴) ≼ 𝐴))
18	8, 17	sylbird 262	. . . . . . 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 416	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦 ∈ 𝑥))
21	20	ssrdv 3942	. . . 4 ⊢ ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
22	21	ex 416	. . 3 ⊢ (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
23	22	rgen 3077	. 2 ⊢ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
24		iscard2 9931	. 2 ⊢ ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
25	1, 23, 24	mpbir2an 721	1 ⊢ (card‘(har‘𝐴)) = (har‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3904   class class class wbr 5099  Oncon0 6342  ‘cfv 6517   ≈ cen 8920   ≼ cdom 8921  harchar 9501  cardccrd 9890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-er 8673  df-en 8924  df-dom 8925  df-oi 9455  df-har 9502  df-card 9894

This theorem is referenced by:  cardprclem  9934  alephcard  10023  pwcfsdom  10538  hargch  10628