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Mirrors > Home > ILE Home > Th. List > carden2bex | GIF version |
Description: If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
carden2bex | ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enen2 6788 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵)) | |
2 | 1 | rabbidv 2701 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵}) |
3 | 2 | inteqd 3814 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵}) |
4 | 3 | adantr 274 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵}) |
5 | cardval3ex 7122 | . . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
6 | 5 | adantl 275 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
7 | entr 6731 | . . . . . 6 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝑥 ≈ 𝐵) | |
8 | 7 | expcom 115 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 → 𝑥 ≈ 𝐵)) |
9 | 8 | reximdv 2558 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∃𝑥 ∈ On 𝑥 ≈ 𝐵)) |
10 | 9 | imp 123 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → ∃𝑥 ∈ On 𝑥 ≈ 𝐵) |
11 | cardval3ex 7122 | . . 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 2200 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∃wrex 2436 {crab 2439 ∩ cint 3809 class class class wbr 3967 Oncon0 4325 ‘cfv 5172 ≈ cen 6685 cardccrd 7116 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-er 6482 df-en 6688 df-card 7117 |
This theorem is referenced by: (None) |
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