Theorem harval2 9949

Description: An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
harval2	⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem harval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		harval 9502	. . . . . . 7 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴})
2	1	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → (har‘𝐴) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴})
3		sdomel 9090	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ≺ 𝑥 → 𝑦 ∈ 𝑥))
4		domsdomtr 9078	. . . . . . . . . . . 12 ⊢ ((𝑦 ≼ 𝐴 ∧ 𝐴 ≺ 𝑥) → 𝑦 ≺ 𝑥)
5	3, 4	impel 513	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑦 ≼ 𝐴 ∧ 𝐴 ≺ 𝑥)) → 𝑦 ∈ 𝑥)
6	5	an4s 670	. . . . . . . . . 10 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → 𝑦 ∈ 𝑥)
7	6	ancoms 462	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) ∧ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)) → 𝑦 ∈ 𝑥)
8	7	3impb 1126	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) ∧ 𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) → 𝑦 ∈ 𝑥)
9	8	rabssdv 4025	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≺ 𝑥) → {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴} ⊆ 𝑥)
10	9	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → {𝑦 ∈ On ∣ 𝑦 ≼ 𝐴} ⊆ 𝑥)
11	2, 10	eqsstrd 3968	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) → (har‘𝐴) ⊆ 𝑥)
12	11	expr 460	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
13	12	ralrimiva 3153	. . 3 ⊢ (𝐴 ∈ dom card → ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
14		ssintrab 4926	. . 3 ⊢ ((har‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ↔ ∀𝑥 ∈ On (𝐴 ≺ 𝑥 → (har‘𝐴) ⊆ 𝑥))
15	13, 14	sylibr 236	. 2 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
16		breq2 5101	. . . 4 ⊢ (𝑥 = (har‘𝐴) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (har‘𝐴)))
17		harcl 9501	. . . . 5 ⊢ (har‘𝐴) ∈ On
18	17	a1i 11	. . . 4 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ On)
19		harsdom 9947	. . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
20	16, 18, 19	elrabd 3651	. . 3 ⊢ (𝐴 ∈ dom card → (har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
21		intss1 4918	. . 3 ⊢ ((har‘𝐴) ∈ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ (har‘𝐴))
22	20, 21	syl 17	. 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥} ⊆ (har‘𝐴))
23	15, 22	eqssd 3951	1 ⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3902  ∩ cint 4902   class class class wbr 5097  dom cdm 5643  Oncon0 6341  ‘cfv 6516   ≼ cdom 8919   ≺ csdm 8920  harchar 9498  cardccrd 9887

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 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

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 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-oi 9452  df-har 9499  df-card 9891

This theorem is referenced by:  harsucnn  9950  alephnbtwn  10021  harval3  44075  aleph1min  44094