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Mirrors > Home > MPE Home > Th. List > cardne | Structured version Visualization version GIF version |
Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.) |
Ref | Expression |
---|---|
cardne | ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6767 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card) | |
2 | cardon 9584 | . . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On | |
3 | 2 | oneli 6338 | . . . . . . . . 9 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On) |
4 | breq1 5070 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵)) | |
5 | 4 | onintss 6280 | . . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴)) |
6 | 3, 5 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ (card‘𝐵) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴)) |
7 | 6 | adantl 485 | . . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴)) |
8 | cardval3 9592 | . . . . . . . . 9 ⊢ (𝐵 ∈ dom card → (card‘𝐵) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵}) | |
9 | 8 | sseq1d 3946 | . . . . . . . 8 ⊢ (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴)) |
10 | 9 | adantr 484 | . . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴)) |
11 | 7, 10 | sylibrd 262 | . . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → (card‘𝐵) ⊆ 𝐴)) |
12 | ontri1 6264 | . . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵))) | |
13 | 2, 3, 12 | sylancr 590 | . . . . . . 7 ⊢ (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵))) |
14 | 13 | adantl 485 | . . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵))) |
15 | 11, 14 | sylibd 242 | . . . . 5 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ¬ 𝐴 ∈ (card‘𝐵))) |
16 | 15 | con2d 136 | . . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)) |
17 | 16 | ex 416 | . . 3 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))) |
18 | 17 | pm2.43d 53 | . 2 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)) |
19 | 1, 18 | mpcom 38 | 1 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 {crab 3066 ⊆ wss 3880 ∩ cint 4873 class class class wbr 5067 dom cdm 5565 Oncon0 6230 ‘cfv 6397 ≈ cen 8643 cardccrd 9575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-ord 6233 df-on 6234 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-fv 6405 df-en 8647 df-card 9579 |
This theorem is referenced by: carden2b 9607 cardlim 9612 cardsdomelir 9613 |
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