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Theorem carden2bex 6718
Description: If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
carden2bex ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = (card‘𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem carden2bex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 enen2 6485 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
21rabbidv 2601 . . . 4 (𝐴𝐵 → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
32inteqd 3667 . . 3 (𝐴𝐵 {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
43adantr 270 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
5 cardval3ex 6714 . . 3 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 271 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
7 entr 6429 . . . . . 6 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87expcom 114 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
98reximdv 2468 . . . 4 (𝐴𝐵 → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ On 𝑥𝐵))
109imp 122 . . 3 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → ∃𝑥 ∈ On 𝑥𝐵)
11 cardval3ex 6714 . . 3 (∃𝑥 ∈ On 𝑥𝐵 → (card‘𝐵) = {𝑦 ∈ On ∣ 𝑦𝐵})
1210, 11syl 14 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐵) = {𝑦 ∈ On ∣ 𝑦𝐵})
134, 6, 123eqtr4d 2125 1 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wrex 2354  {crab 2357   cint 3662   class class class wbr 3811  Oncon0 4153  cfv 4967  cen 6383  cardccrd 6708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-er 6220  df-en 6386  df-card 6709
This theorem is referenced by: (None)
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