Theorem carden2bex 7249

Description: If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
carden2bex	⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem carden2bex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enen2 6897	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
2	1	rabbidv 2749	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
3	2	inteqd 3875	. . 3 ⊢ (𝐴 ≈ 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
4	3	adantr 276	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
5		cardval3ex 7245	. . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
6	5	adantl 277	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
7		entr 6838	. . . . . 6 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝑥 ≈ 𝐵)
8	7	expcom 116	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 → 𝑥 ≈ 𝐵))
9	8	reximdv 2595	. . . 4 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∃𝑥 ∈ On 𝑥 ≈ 𝐵))
10	9	imp 124	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → ∃𝑥 ∈ On 𝑥 ≈ 𝐵)
11		cardval3ex 7245	. . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐵 → (card‘𝐵) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
12	10, 11	syl 14	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐵) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
13	4, 6, 12	3eqtr4d 2236	1 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wrex 2473  {crab 2476  ∩ cint 3870   class class class wbr 4029  Oncon0 4394  ‘cfv 5254   ≈ cen 6792  cardccrd 7239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-er 6587  df-en 6795  df-card 7240

This theorem is referenced by: (None)