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Mirrors > Home > ILE Home > Th. List > cardonle | GIF version |
Description: The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
cardonle | ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oncardval 7042 | . 2 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) | |
2 | enrefg 6658 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴) | |
3 | breq1 3932 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴)) | |
4 | 3 | intminss 3796 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴) |
5 | 2, 4 | mpdan 417 | . 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴) |
6 | 1, 5 | eqsstrd 3133 | 1 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 {crab 2420 ⊆ wss 3071 ∩ cint 3771 class class class wbr 3929 Oncon0 4285 ‘cfv 5123 ≈ cen 6632 cardccrd 7035 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-en 6635 df-card 7036 |
This theorem is referenced by: card0 7044 |
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