Theorem harcard 9902

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 9476	. 2 ⊢ (har‘𝐴) ∈ On
2		harndom 9479	. . . . . . 7 ⊢ ¬ (har‘𝐴) ≼ 𝐴
3		simpll 767	. . . . . . . . 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 9478	. . . . . . . . . . 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 6359	. . . . . . . . 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 776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥 ⊆ 𝑦) → (har‘𝐴) ≈ 𝑥)
11		ssdomg 8949	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦))
12	11	elv 3447	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑦 → 𝑥 ≼ 𝑦)
13	6	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ≼ 𝐴)
14		domtr 8956	. . . . . . . . . . 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 8961	. . . . . . . . . 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 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 3941	. . . 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 3054	. 2 ⊢ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
25		iscard2 9900	. 2 ⊢ ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
26	1, 24, 25	mpbir2an 712	1 ⊢ (card‘(har‘𝐴)) = (har‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ⊆ wss 3903   class class class wbr 5100  Oncon0 6325  ‘cfv 6500   ≈ cen 8892   ≼ cdom 8893  harchar 9473  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-er 8645  df-en 8896  df-dom 8897  df-oi 9427  df-har 9474  df-card 9863

This theorem is referenced by:  cardprclem  9903  alephcard  9992  pwcfsdom  10506  hargch  10596