Theorem unveldom 30818

Description: The universe is an element of the domain of the probability, the universe (entire probability space) being ∪ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)

Assertion

Ref	Expression
unveldom	⊢ (𝑃 ∈ Prob → ∪ dom 𝑃 ∈ dom 𝑃)

Proof of Theorem unveldom

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ Prob)
2	1	unveldomd 30817	1 ⊢ (𝑃 ∈ Prob → ∪ dom 𝑃 ∈ dom 𝑃)

Syntax hints:   → wi 4   ∈ wcel 2145  ∪ cuni 4575  dom cdm 5250  Probcprb 30809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-esum 30430  df-siga 30511  df-meas 30599  df-prob 30810

This theorem is referenced by: (None)