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Theorem carden2bex 7190
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 6843 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
21rabbidv 2728 . . . 4 (𝐴𝐵 → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
32inteqd 3851 . . 3 (𝐴𝐵 {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
43adantr 276 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → {𝑦 ∈ On ∣ 𝑦𝐴} = {𝑦 ∈ On ∣ 𝑦𝐵})
5 cardval3ex 7186 . . 3 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 277 . 2 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
7 entr 6786 . . . . . 6 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87expcom 116 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
98reximdv 2578 . . . 4 (𝐴𝐵 → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ On 𝑥𝐵))
109imp 124 . . 3 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → ∃𝑥 ∈ On 𝑥𝐵)
11 cardval3ex 7186 . . 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 2220 1 ((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wrex 2456  {crab 2459   cint 3846   class class class wbr 4005  Oncon0 4365  cfv 5218  cen 6740  cardccrd 7180
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-er 6537  df-en 6743  df-card 7181
This theorem is referenced by: (None)
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