Theorem carden2bex 6369

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 6312	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
2	1	rabbidv 2549	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
3	2	inteqd 3620	. . 3 ⊢ (𝐴 ≈ 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
4	3	adantr 261	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
5		cardval3ex 6365	. . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
6	5	adantl 262	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
7		entr 6264	. . . . . 6 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝑥 ≈ 𝐵)
8	7	expcom 109	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 → 𝑥 ≈ 𝐵))
9	8	reximdv 2420	. . . 4 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∃𝑥 ∈ On 𝑥 ≈ 𝐵))
10	9	imp 115	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → ∃𝑥 ∈ On 𝑥 ≈ 𝐵)
11		cardval3ex 6365	. . 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 2082	1 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wrex 2307  {crab 2310  ∩ cint 3615   class class class wbr 3764  Oncon0 4100  ‘cfv 4902   ≈ cen 6219  cardccrd 6359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-er 6106  df-en 6222  df-card 6360

This theorem is referenced by: (None)