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Mirrors > Home > MPE Home > Th. List > carden2a | Structured version Visualization version GIF version |
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 9964 are meant to replace carden 10548 in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013.) |
Ref | Expression |
---|---|
carden2a | ⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2935 | . 2 ⊢ ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅) | |
2 | ndmfv 6920 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅) | |
3 | eqeq1 2730 | . . . . . . 7 ⊢ ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅)) | |
4 | 2, 3 | imbitrrid 245 | . . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ 𝐵 ∈ dom card → (card‘𝐴) = ∅)) |
5 | 4 | con1d 145 | . . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → 𝐵 ∈ dom card)) |
6 | 5 | imp 406 | . . . 4 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card) |
7 | cardid2 9950 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵) |
9 | breq2 5145 | . . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵))) | |
10 | entr 9004 | . . . . . 6 ⊢ ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≈ 𝐵) | |
11 | 10 | ex 412 | . . . . 5 ⊢ (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)) |
12 | 9, 11 | biimtrdi 252 | . . . 4 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))) |
13 | cardid2 9950 | . . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
14 | ndmfv 6920 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
15 | 13, 14 | nsyl4 158 | . . . . 5 ⊢ (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴) |
16 | 15 | ensymd 9003 | . . . 4 ⊢ (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴)) |
17 | 12, 16 | impel 505 | . . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)) |
18 | 8, 17 | mpd 15 | . 2 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐴 ≈ 𝐵) |
19 | 1, 18 | sylan2b 593 | 1 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 class class class wbr 5141 dom cdm 5669 ‘cfv 6537 ≈ cen 8938 cardccrd 9932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-card 9936 |
This theorem is referenced by: card1 9965 |
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