Theorem card2on 9243

Description: The alternate definition of the cardinal of a set given in cardval2 9680 always gives a set, and indeed an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)

Assertion

Ref	Expression
card2on	⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On

Distinct variable group:   𝑥,𝐴

Proof of Theorem card2on

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

Step	Hyp	Ref	Expression
1		onelon 6276	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
2		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
3		onelss 6293	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 8741	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (𝑦 ⊆ 𝑧 → 𝑦 ≼ 𝑧))
6	2, 4, 5	mpsyl 68	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ≼ 𝑧)
7	1, 6	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝑧))
8		domsdomtr 8848	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴) → 𝑦 ≺ 𝐴)
9	8	anim2i 616	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ (𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
10	9	anassrs 467	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝑧) ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
11	7, 10	sylan 579	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
12	11	exp31 419	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → (𝑧 ≺ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))))
13	12	com12 32	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ On → (𝑧 ≺ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))))
14	13	impd 410	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑧 → ((𝑧 ∈ On ∧ 𝑧 ≺ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴)))
15		breq1 5073	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
16	15	elrab 3617	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≺ 𝐴))
17		breq1 5073	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝐴 ↔ 𝑦 ≺ 𝐴))
18	17	elrab 3617	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ 𝐴))
19	14, 16, 18	3imtr4g 295	. . . . . . 7 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
20	19	imp 406	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
21	20	gen2 1800	. . . . 5 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
22		dftr2 5189	. . . . 5 ⊢ (Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
23	21, 22	mpbir 230	. . . 4 ⊢ Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
24		ssrab2 4009	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ On
25		ordon 7604	. . . 4 ⊢ Ord On
26		trssord 6268	. . . 4 ⊢ ((Tr {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
27	23, 24, 25, 26	mp3an 1459	. . 3 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}
28		hartogs 9233	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)
29		sdomdom 8723	. . . . . . 7 ⊢ (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴)
30	29	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → (𝑥 ≺ 𝐴 → 𝑥 ≼ 𝐴))
31	30	ss2rabi 4006	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
32		ssexg 5242	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V)
33	31, 32	mpan 686	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V)
34		elong 6259	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
35	28, 33, 34	3syl 18	. . 3 ⊢ (𝐴 ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}))
36	27, 35	mpbiri 257	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
37		0elon 6304	. . . 4 ⊢ ∅ ∈ On
38		eleq1 2826	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On ↔ ∅ ∈ On))
39	37, 38	mpbiri 257	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
40		df-ne 2943	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ≠ ∅ ↔ ¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅)
41		rabn0 4316	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≺ 𝐴)
42	40, 41	bitr3i 276	. . . 4 ⊢ (¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ ↔ ∃𝑥 ∈ On 𝑥 ≺ 𝐴)
43		relsdom 8698	. . . . . 6 ⊢ Rel ≺
44	43	brrelex2i 5635	. . . . 5 ⊢ (𝑥 ≺ 𝐴 → 𝐴 ∈ V)
45	44	rexlimivw 3210	. . . 4 ⊢ (∃𝑥 ∈ On 𝑥 ≺ 𝐴 → 𝐴 ∈ V)
46	42, 45	sylbi 216	. . 3 ⊢ (¬ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = ∅ → 𝐴 ∈ V)
47	39, 46	nsyl4 158	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On)
48	36, 47	pm2.61i 182	1 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253   class class class wbr 5070  Tr wtr 5187  Ord word 6250  Oncon0 6251   ≼ cdom 8689   ≺ csdm 8690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-oi 9199

This theorem is referenced by: (None)