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Theorem carden2bex 6880
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 6613 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
21rabbidv 2611 . . . 4 (𝐴𝐵 → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
32inteqd 3701 . . 3 (𝐴𝐵 {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
43adantr 271 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
5 cardval3ex 6876 . . 3 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 272 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
7 entr 6557 . . . . . 6 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87expcom 115 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
98reximdv 2475 . . . 4 (𝐴𝐵 → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ On 𝑥𝐵))
109imp 123 . . 3 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → ∃𝑥 ∈ On 𝑥𝐵)
11 cardval3ex 6876 . . 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 2131 1 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  wrex 2361  {crab 2364   cint 3696   class class class wbr 3853  Oncon0 4201  cfv 5030  cen 6511  cardccrd 6870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-er 6308  df-en 6514  df-card 6871
This theorem is referenced by: (None)
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