Theorem carden2bex 7256

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 6902	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
2	1	rabbidv 2752	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
3	2	inteqd 3879	. . 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 7252	. . 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 6843	. . . . . 6 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝑥 ≈ 𝐵)
8	7	expcom 116	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 → 𝑥 ≈ 𝐵))
9	8	reximdv 2598	. . . 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 7252	. . 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 2239	1 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wrex 2476  {crab 2479  ∩ cint 3874   class class class wbr 4033  Oncon0 4398  ‘cfv 5258   ≈ cen 6797  cardccrd 7246

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-er 6592  df-en 6800  df-card 7247

This theorem is referenced by: (None)