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Theorem carden2bex 7166
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 6819 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
21rabbidv 2719 . . . 4 (𝐴𝐵 → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
32inteqd 3836 . . 3 (𝐴𝐵 {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
43adantr 274 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
5 cardval3ex 7162 . . 3 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 275 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
7 entr 6762 . . . . . 6 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87expcom 115 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
98reximdv 2571 . . . 4 (𝐴𝐵 → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ On 𝑥𝐵))
109imp 123 . . 3 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → ∃𝑥 ∈ On 𝑥𝐵)
11 cardval3ex 7162 . . 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 2213 1 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wrex 2449  {crab 2452   cint 3831   class class class wbr 3989  Oncon0 4348  cfv 5198  cen 6716  cardccrd 7156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-card 7157
This theorem is referenced by: (None)
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