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Theorem omssrncard 43530
Description: All natural numbers are cardinals. (Contributed by RP, 1-Oct-2023.)
Assertion
Ref Expression
omssrncard ω ⊆ ran card

Proof of Theorem omssrncard
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnon 7893 . . 3 (𝑥 ∈ ω → 𝑥 ∈ On)
2 onelon 6411 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3 simpl 482 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑥 ∈ On)
4 simpr 484 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
5 onelpss 6426 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥)))
65biimpa 476 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → (𝑦𝑥𝑦𝑥))
72, 3, 4, 6syl21anc 838 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦𝑥𝑦𝑥))
8 df-pss 3983 . . . . . . . . . 10 (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥))
97, 8sylibr 234 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
109ex 412 . . . . . . . 8 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
111, 10syl 17 . . . . . . 7 (𝑥 ∈ ω → (𝑦𝑥𝑦𝑥))
1211imdistani 568 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦𝑥) → (𝑥 ∈ ω ∧ 𝑦𝑥))
13 php 9245 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦𝑥) → ¬ 𝑥𝑦)
1412, 13syl 17 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦𝑥) → ¬ 𝑥𝑦)
15 ensymb 9041 . . . . 5 (𝑥𝑦𝑦𝑥)
1614, 15sylnib 328 . . . 4 ((𝑥 ∈ ω ∧ 𝑦𝑥) → ¬ 𝑦𝑥)
1716ralrimiva 3144 . . 3 (𝑥 ∈ ω → ∀𝑦𝑥 ¬ 𝑦𝑥)
18 elrncard 43527 . . 3 (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 ¬ 𝑦𝑥))
191, 17, 18sylanbrc 583 . 2 (𝑥 ∈ ω → 𝑥 ∈ ran card)
2019ssriv 3999 1 ω ⊆ ran card
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2106  wne 2938  wral 3059  wss 3963  wpss 3964   class class class wbr 5148  ran crn 5690  Oncon0 6386  ωcom 7887  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-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-lim 6391  df-suc 6392  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-om 7888  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977
This theorem is referenced by:  0iscard  43531  1iscard  43532  nna1iscard  43535
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