Theorem unveldom 35159

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 23	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ Prob)
2	1	unveldomd 35158	1 ⊢ (𝑃 ∈ Prob → ∪ dom 𝑃 ∈ dom 𝑃)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∪ cuni 4872  dom cdm 5658  Probcprb 35150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-esum 34771  df-siga 34852  df-meas 34940  df-prob 35151

This theorem is used by: (None)