Theorem carden2bex 7399

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 7032	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
2	1	rabbidv 2790	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
3	2	inteqd 3934	. . 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 7394	. . 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 6963	. . . . . 6 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝑥 ≈ 𝐵)
8	7	expcom 116	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 → 𝑥 ≈ 𝐵))
9	8	reximdv 2632	. . . 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 7394	. . 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 2273	1 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wrex 2510  {crab 2513  ∩ cint 3929   class class class wbr 4089  Oncon0 4462  ‘cfv 5328   ≈ cen 6912  cardccrd 7386

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-er 6707  df-en 6915  df-card 7388

This theorem is referenced by: (None)