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Mirrors > Home > MPE Home > Th. List > cardonle | Structured version Visualization version 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 9993 | . 2 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) | |
2 | enrefg 9023 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴) | |
3 | breq1 5151 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴)) | |
4 | 3 | intminss 4979 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴) |
5 | 2, 4 | mpdan 687 | . 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ⊆ 𝐴) |
6 | 1, 5 | eqsstrd 4034 | 1 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 ∩ cint 4951 class class class wbr 5148 Oncon0 6386 ‘cfv 6563 ≈ cen 8981 cardccrd 9973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-en 8985 df-card 9977 |
This theorem is referenced by: card0 9996 iscard 10013 iscard2 10014 carduni 10019 cardom 10024 alephinit 10133 cfle 10292 cfflb 10297 pwfseqlem5 10701 harval3 43528 |
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